\documentclass[12pt,reqno]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{epsf} \usepackage{float} \usepackage[usenames]{color} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Examle} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newenvironment{proof}{\textit{Proof:} \ignorespaces}{\nopagebreak\null\hfill\hbox{$\Box$}} \newcommand{\cnd}{\hbox{\hspace*{-0.6ex}% \hbox{\raisebox{-0.6ex}{$\Box$}}}\vspace{0.5cm}\par} \newcommand{\cndd}{\mbox{\raisebox{-0.6ex}{$\Box$} \vspace{0.5cm}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \mbox{\null} \hspace*{4cm} \end{center} \begin{center} \vskip 1cm{\LARGE\bf New Ramanujan-Type Formulas and \\ \vskip .1in Quasi-Fibonacci Numbers of Order 7} \vskip 1cm \large Roman Witu{\l}a and Damian S{\l}ota \\ Institute of Mathematics \\ Silesian University of Technology \\ Kaszubska 23 \\ Gliwice 44-100 \\ Poland \\ \href{mailto:r.witula@polsl.pl}{\tt r.witula@polsl.pl} \\ \href{mailto:d.slota@polsl.pl}{\tt d.slota@polsl.pl} \\ \end{center} \vskip .2in \begin{abstract} We give applications of the quasi-Fibonacci numbers of order $7$ and the so-called sine-Fibonacci numbers of order $7$ and many other new kinds of recurrent sequences to the decompositions of some polynomials. We also present the characteristic equations, generating functions and some properties of all these sequences. Finally, some new Ramanujan-type formulas are generated. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction}\label{roz1} The scope of the paper is the generalization of the following decompositions of polynomials \cite{GrzymkowskiWitula,Ireland,Shevelev,surowski,Watkins}: \begin{align} \big( \mathbb{X} - &2 \sin (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 2 \sin (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 2 \sin (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3 -\sqrt{7}\, \mathbb{X}^2 +\sqrt{7},\label{w1-a}\\ \big( \mathbb{X} - &4 \sin^2 (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 4 \sin^2 (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 4 \sin^2 (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3 - 7\, \mathbb{X}^2 + 14\, \mathbb{X} - 7,\label{w1-b}\\ \big( \mathbb{X} - &8 \sin^3 (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 8 \sin^3 (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 8 \sin^3 (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3-4\, \sqrt{7}\, \mathbb{X}^2 +21\, \mathbb{X} +7\, \sqrt{7} ,\label{w1-c}\\ \big( \mathbb{X} - &2 \cos (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 2 \cos (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 2 \cos (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3 + \mathbb{X}^2 - 2\, \mathbb{X} -1,\label{w1-d}\\ \big( \mathbb{X} - &4 \cos^2 (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 4 \cos^2 (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 4 \cos^2 (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3 - 5\, \mathbb{X}^2 + 6\, \mathbb{X} - 1,\label{w1-e}\\ \big( \mathbb{X} - &8 \cos^3 (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 8 \cos^3 (\tfrac{4\,\pi}{7})\big) \big( \mathbb{X} - 8 \cos^3 (\tfrac{8\,\pi}{7})\big) =\mathbb{X}^3+4\, \mathbb{X}^2-11\, \mathbb{X}-1,\label{w1-f}\\ \big( \mathbb{X} - &8 \sin (\tfrac{2\,\pi}{7})\, \cos^2 (\tfrac{8\,\pi}{7})\big) \big( \mathbb{X} - 8 \sin (\tfrac{4\,\pi}{7})\, \cos^2 (\tfrac{2\,\pi}{7})\big) \big( \mathbb{X} - 8 \sin (\tfrac{8\,\pi}{7})\, \cos^2 (\tfrac{4\,\pi}{7})\big)=\nonumber\\ &=\mathbb{X}^3 - 3\, \sqrt{7}\, \mathbb{X}^2 + 14\, \mathbb{X} + \sqrt{7},\label{w1-g}\\ etc.& \nonumber \end{align} The main incentive for generating the decompositions of these polynomials is provided by the properties of the so-called quasi-Fibonacci numbers of order $7$, $A_{n}(\delta)$, $B_{n}(\delta)$ and $C_{n}(\delta)$, $n\in \mathbb{N}$, described in~\cite{wsw1} by means of the relations \begin{equation} (1+\delta\, (\xi^{k}+\xi^{6k}))^n = A_n(\delta)+B_n(\delta)\, (\xi^{k}+\xi^{6k})+C_n(\delta)\, (\xi^{2k}+\xi^{5k}) \end{equation} for $k=1,2,3$, where $\xi\in\mathbb{C}$ is a~primitive root of unity of order $7$ (i.e.,~$\xi^7=1$ and $\xi\neq 1$), $\delta\in \mathbb{C}$, $\delta\neq 0$. Besides, an essential r{\^{o}}le in the decompositions of polynomials discussed in the paper is played by related numbers ($\delta\in \mathbb{C}$, $n\in\mathbb{N}$): \begin{multline}\label{w-and} \mathcal{A}_{n}(\delta) := 3\, A_{n}(\delta) - B_{n}(\delta) -C_{n}(\delta) = \\ =\big(1+\delta\, (\xi+\xi^6)\big)^n +\big(1+\delta\, (\xi^2+\xi^5)\big)^n +\big(1+\delta\, (\xi^3+\xi^4)\big)^n \end{multline} and \begin{align} \mathcal{B}_{n}(\delta) &:= \frac{1}{2} \big( \big(\mathcal{A}_{n}(\delta)\big)^{2} - \mathcal{A}_{2n}(\delta) \big) = \nonumber\\ &=\Big( \big( 1 + 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big) \big( 1 + 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big)\Big)^{n}+{}\nonumber\\ &\phantom{=}\hspace*{2ex} {} + \Big( \big( 1 + 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big) \big( 1 + 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)\Big)^{n}+{}\nonumber\\ &\phantom{=}\hspace*{2ex} {} + \Big( \big( 1 + 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big) \big( 1 + 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)\Big)^{n}={}\nonumber\\ &= 3\,\big(A_{n}(\delta)\big)^{2} - 2\, A_{n}(\delta)\, B_{n}(\delta) - 2\, A_{n}(\delta)\, C_{n}(\delta) + {} \nonumber \\ &\phantom{=}\hspace*{2ex} {} + 3\, B_{n}(\delta)\, C_{n}(\delta) -2\,\big(B_{n}(\delta)\big)^{2} -2\,\big(C_{n}(\delta)\big)^{2} = {} \nonumber\\ &=A_{n}(\delta)\, \big( 2\, \mathcal{A}_{n}(\delta) - 3 A_{n}(\delta) \Big) - 2\, \big( B_{n}(\delta) - C_{n}(\delta) \Big)^{2} - B_{n}(\delta)\, C_{n}(\delta).\label{w-bnd} \end{align} Furthermore, to simplify the formulas, we will write \begin{equation} \mathcal{A}_{n}=\mathcal{A}_{n}(1),\ \ \mathcal{B}_{n}=\mathcal{B}_{n}(1),\ \ A_{n}=A_{n}(1),\ \ B_{n}=B_{n}(1)\ \ \mbox{and}\ \ C_{n}=C_{n}(1), \end{equation} for every $n\in \mathbb{N}$. We note that the tables of values of these numbers can be found in the article~\cite{wsw1}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Basic decompositions}\label{roz2a} Witu{\l}a et al.\ \cite{wsw1} determined the following two formulas: \begin{multline} \big( \mathbb{X} - \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - \mathcal{B}_{n}\, \mathbb{X}^2 + (-1)^{n}\, \mathcal{A}_{n} \, \mathbb{X} - 1\label{wzor-gw1new} \end{multline} and \begin{multline}\label{w1.10} \big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - \mathcal{A}_{n} (\delta)\, \mathbb{X}^2 + \mathcal{B}_{n}(\delta) \, \mathbb{X} - \big(1-\delta-2\, \delta^2+\delta^3\big)^n. \end{multline} From~(\ref{w1.10}) three special formulas follow: \begin{multline}\label{wiel-114} \big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - \big( \tfrac{3}{2}\big)^{n}\, \mathcal{A}_{n} \big(\tfrac{1}{6}\big)\, \mathbb{X}^2 + \big( \tfrac{3}{2}\big)^{2n}\, \mathcal{B}_{n} \big(\tfrac{1}{6}\big)\, \mathbb{X} - \big( \tfrac{13}{8}\big)^{2n}, \end{multline} \begin{multline} \big( \mathbb{X} - \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{2n} \big) \big( \mathbb{X} - \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{2n} \big) \big( \mathbb{X} - \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{2n} \big) = {}\\ {} = \mathbb{X}^3 - 2^{n}\, \mathcal{A}_{n} \big(-\tfrac{1}{2}\big)\, \mathbb{X}^2 + 2^{2n}\, \mathcal{B}_{n} \big(-\tfrac{1}{2}\big)\, \mathbb{X} - 7^n \end{multline} and \begin{multline}\label{wz116} \big( \mathbb{X} - \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{2n} \big) \big( \mathbb{X} - \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \big) \big( \mathbb{X} - \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} \big) = {}\\ {} = \mathbb{X}^3 - 2^{n}\, \mathcal{A}_{n} \big(\tfrac{1}{2}\big)\, \mathbb{X}^2 + 2^{2n}\, \mathcal{B}_{n} \big(\tfrac{1}{2}\big)\, \mathbb{X} - 1. \end{multline} Comparing formulas~(\ref{wzor-gw1new}) and~(\ref{wz116}) two new identities are generated \begin{equation} \mathcal{A}_{2n}=2^{2n}\, \mathcal{B}_{n} \big(\tfrac{1}{2}\big) \quad \mbox{ and } \quad \mathcal{B}_{2n}=2^{n}\, \mathcal{A}_{n} \big(\tfrac{1}{2}\big). \end{equation} We also have the decomposition \begin{multline}\label{w1.19} \Big( \mathbb{X} - \big( \big(1+\delta\, (\xi+\xi^6)\big)\, \big(1+\delta\, (\xi^2+\xi^5)\big)\big)^{n} \Big)\, \Big( \mathbb{X} - \big( \big(1+\delta\, (\xi+\xi^6)\big)\, \big(1+\delta\, (\xi^3+\xi^4)\big)\big)^{n} \Big)\times{}\\ {}\times \Big( \mathbb{X} - \big( \big(1+\delta\, (\xi^2+\xi^5)\big)\, \big(1+\delta\, (\xi^3+\xi^4)\big)\big)^{n} \Big) = {}\\ {} = \Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{2\, \pi}{7} ) \big)\, \big(1+2\,\delta\, \cos (\tfrac{4\, \pi}{7} ) \big) \big)^{n} \Big)\, \Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{2\, \pi}{7} ) \big)\, \big(1+2\,\delta\, \cos (\tfrac{6\, \pi}{7} ) \big) \big)^{n} \Big)\times {} \\ {}\times \Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{4\, \pi}{7} ) \big)\, \big(1+2\,\delta\, \cos (\tfrac{6\, \pi}{7} ) \big) \big)^{n} \Big) ={}\\ {}= \mathbb{X}^{3} - \mathcal{B}_{n}(\delta)\, \mathbb{X}^{2} + \big(1-\delta-2\, \delta^2+\delta^3\big)\, \mathcal{A}_{n}(\delta)\, \mathbb{X} - \big(1-\delta-2\, \delta^2+\delta^3\big)^{2n} = {} \\ {} := r_{n}(\mathbb{X}; \delta). \end{multline} Now let us set \begin{align} \Xi_{n} &:= \Xi_{n} (\delta, \varepsilon, \eta ) = 2^n\, \Big( \delta\, \cos^n \big(\tfrac{2\, \pi}{7}\big) + \varepsilon\, \cos^n \big(\tfrac{4\, \pi}{7}\big) + \eta\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big),\label{w2.8}\\ \Upsilon_{n} &:= \Upsilon_{n} (\delta, \varepsilon, \eta ) = 2^n\, \Big( \varepsilon\, \cos^n \big(\tfrac{2\, \pi}{7}\big) + \eta\, \cos^n \big(\tfrac{4\, \pi}{7}\big) + \delta\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big),\\ \Theta_{n} &:= \Theta_{n} (\delta, \varepsilon, \eta ) = 2^n\, \Big( \eta\, \cos^n \big(\tfrac{2\, \pi}{7}\big) + \delta\, \cos^n \big(\tfrac{4\, \pi}{7}\big) + \varepsilon\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big) \end{align} for any $\delta, \varepsilon, \eta \in \mathbb{C}$ and $n\in \mathbb{N}_0$. \begin{lemma}\label{lem2.1} The following general decomposition formula holds \begin{multline}\label{choinka} (\mathbb{X} -\Xi_{n}) (\mathbb{X} -\Upsilon_{n}) (\mathbb{X} -\Theta_{n}) =\\ = \mathbb{X}^3 - (\delta+\varepsilon+\eta)\, \mathcal{A}_{n}\, \mathbb{X}^2 +\tfrac{1}{2}\, (-1)^n\, \big( (\delta+\varepsilon)^2 +(\delta+\eta)^2 +(\varepsilon+\eta)^2 \big)\, \mathcal{A}_{n}\, \mathbb{X} - {}\\ {} - \Big[ \delta\,\varepsilon\,\eta\, (\mathcal{B}_{3n}+3) + \delta^3 + \varepsilon^3 + \eta^3 +(-1)^n\, \big( \delta\,\varepsilon^2 + \varepsilon\, \eta^2 + \eta\, \delta^2 \big)\, \mathcal{A}_{n}(-1) + {}\\ {}+ 2^{2n}\, \big( \delta^2\,\varepsilon + \varepsilon^2\, \eta + \eta^2\, \delta \big)\, \sum_{k=0}^{n} (-1)^k\, \binom{n}{k}\, \mathcal{A}_{2n-k} \big(\tfrac{1}{2}\big) \Big]. \end{multline} \end{lemma} Below an illustrative example connected with Lemma~\ref{lem2.1} will be presented. \begin{example}\label{example2.2} {\rm A.~M.~Yaglom and I.~M.~Yaglom~\cite{Yaglom} (see also~\cite[problem~230]{ShklyarskyUSA} and~\cite[problem~329]{ShklyarskyMir}) considered the following polynomial: $$ w(x)=\binom{2m + 1}{1}\, x^m - \binom{2m + 1}{3}\, x^{m-1} + \binom{2m + 1}{5}\, x^{m-2} - \ldots $$ and proved that it has the roots $$ x_k = \cot^2 \Big( \frac{k \, \pi}{2m + 1} \Big), \qquad k = 1, 2, \ldots, m. $$ In particular, for $m = 3$ taking into account that $$ \cot^2 \big( \tfrac{\pi}{7} \big) = \cot^2 \big( \tfrac{8\, \pi}{7} \big), \qquad \cot^2 \big( \tfrac{3\,\pi}{7} \big) = \cot^2 \big( \tfrac{4\, \pi}{7} \big) $$ we have the decomposition (see also formula~(\ref{w6.14}) below): $$ \big( x - \cot^2 \big( \tfrac{2\,\pi}{7} \big) \big) \, \big( x - \cot^2 \big( \tfrac{4\,\pi}{7} \big) \big) \, \big( x - \cot^2 \big( \tfrac{8\,\pi}{7} \big) \big) = x^3 - 5\, x^2 + 3\, x - \tfrac{1}{7} $$ and as a corollary~-- the decomposition \begin{equation}\label{ex-w5} \big( x + 7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) \big) \, \big( x + 7\, \cot^2 \big( \tfrac{4\,\pi}{7} \big) \big) \, \big( x + 7\, \cot^2 \big( \tfrac{8\,\pi}{7} \big) \big) = x^3 + 35\, x^2 + 147\, x +49. \end{equation} According to~(\ref{w2.8}) for $n = 1$, let us try to find a~linear combination $$ -7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) = 2\,\Big( \delta\, \cos \big(\tfrac{2\, \pi}{7}\big) + \varepsilon\, \cos \big(\tfrac{4\, \pi}{7}\big) + \eta\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big) $$ or, the same, \begin{multline*} -7\, \cos^2 \big( \tfrac{2\,\pi}{7} \big) = 2\,\Big( \delta\, \cos \big(\tfrac{2\, \pi}{7}\big) + \varepsilon\, \cos \big(\tfrac{4\, \pi}{7}\big) + \eta\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big) - {}\\ {}- \delta\, \cos^3 \big(\tfrac{2\, \pi}{7}\big) -\varepsilon\, \cos \big(\tfrac{4\, \pi}{7}\big)\, \cos^2 \big(\tfrac{2\, \pi}{7}\big) -\eta\, \cos \big(\tfrac{8\, \pi}{7}\big)\, \cos^2 \big(\tfrac{2\, \pi}{7}\big). \end{multline*} Decreasing powers and taking into account the identity $$ \cos \big(\tfrac{2\, \pi}{7}\big) + \cos \big(\tfrac{4\, \pi}{7}\big) + \cos \big(\tfrac{8\, \pi}{7}\big) = -\frac{1}{2} $$ we find $$ \big( 2\, \delta+ \varepsilon -3\, \eta \big) \cos \big(\tfrac{2\, \pi}{7}\big) + \big( \delta+ 3\,\varepsilon -3\, \eta +7 \big) \cos \big(\tfrac{4\, \pi}{7}\big) = -\frac{\delta}{2} +\frac{\varepsilon}{2} +\eta-7. $$ Thus, for finding $\delta$, $\varepsilon$, $\eta$ we have the linear system (see Corollary~2.5 in~\cite{wsw1}): $$ \left\{ \begin{array}{l} 2\, \delta+ \varepsilon -3\, \eta = 0 \\ \delta+ 3\,\varepsilon -3\, \eta = -7 \\ \delta - \varepsilon - 2\, \eta = -14 \end{array} \right. $$ with the solution $$ \delta=17,\qquad \varepsilon=5,\qquad \eta=13. $$ Hence, $$ -7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) = 2\,\Big( 17\, \cos \big(\tfrac{2\, \pi}{7}\big) + 5\, \cos \big(\tfrac{4\, \pi}{7}\big) + 13\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big). $$ Analogously, we obtain \begin{align*} -7\, \cot^2 \big( \tfrac{4\,\pi}{7} \big) &= 2\,\Big( 13\, \cos \big(\tfrac{2\, \pi}{7}\big) + 17\, \cos \big(\tfrac{4\, \pi}{7}\big) + 5\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big),\\ -7\, \cot^2 \big( \tfrac{8\,\pi}{7} \big) &= 2\,\Big( 5\, \cos \big(\tfrac{2\, \pi}{7}\big) + 13\, \cos \big(\tfrac{4\, \pi}{7}\big) + 17\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big). \end{align*} Thus the decomposition~(\ref{ex-w5}) corresponds to Lemma~\ref{lem2.1} with $$ \Xi_{1}(17, 5, 13),\quad \Upsilon_{1}(17, 5, 13),\quad \Theta_{1}(17, 5, 13). $$ } \end{example} \begin{remark} {\rm The identity~(\ref{ex-w5}) was found earlier by Shevelev~\cite{Shevelev}. } \end{remark} \begin{remark} {\rm The formula~(\ref{choinka}), in some cases which are subject of our interest, especially when coefficients $\delta$, $\varepsilon$, $\eta$ are the corresponding values of trigonometric functions, becomes rather complicated. Thus, in the next two sections we attempt to designate the relevant coefficients of decomposition~(\ref{choinka}), including the recurrent coefficients, on the grounds of new sequences that are easier to analyze. } \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The first group of special cases of~(\ref{choinka})} Let us set \begin{align} a_{n} &= 2^{2n+1}\, \Big[ \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} + \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.1}\\ b_{n} &= 2^{2n+1}\, \Big[ \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} + \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.2}\\ c_{n} &= 2^{2n+1}\, \Big[ \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} + \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.3} \end{align} for $n=0,1,2,\ldots$ \begin{lemma}\label{f2s-lem1} The following recurrence relations hold: \begin{equation}\label{f2s-w1g} \left\{ \begin{array}{l} a_{n+1} = 2\, a_{n} + b_{n},\\ b_{n+1} = a_{n} + 2\, b_{n} - c_{n},\\ c_{n+1} = c_{n} - b_{n},\\ \end{array} \right. \end{equation} for $n=0,1,2,\ldots$ and $a_{0}=b_{0}=c_{0}=\sqrt{7}$. Moreover, elements of each sequences $\{a_n\}_{n=0}^{\infty}$, $\{b_n\}_{n=0}^{\infty}$ and $\{c_n\}_{n=0}^{\infty}$ satisfy the recurrence equation \begin{equation}\label{f2s-w2g} x_{n+2} - 5\, x_{n+1} + 6\, x_{n} - x_{n-1} = 0, \qquad n=0,1,2,\ldots \end{equation} (in view of decomposition~(\ref{w1-e}) an appropriate characteristic polynomial is compatible with the definition of numbers $a_n$, $b_n$ and~$c_n$). \end{lemma} The first twelve values of numbers $a_{n}^{*}=a_{n}/\sqrt{7}$, $b_{n}^{*}=b_{n}/\sqrt{7}$ and $c_{n}^{*}=c_{n}/\sqrt{7}$ are presented in Table~\ref{f2s-tab1}. Now, let us set \begin{align} \alpha_{n} &= 2^{2n}\, \Big[ \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big],\\ \beta_{n} &= 2^{2n}\, \Big[ \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big],\\ \gamma_{n} &= 2^{2n}\, \Big[ \sin (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} + \sin (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big], \end{align} for $n=1,2,\ldots$ \begin{lemma}\label{f2s-lem2} We have $$ \alpha_1=0,\qquad \beta_1=-2\, \sqrt{7},\qquad \mbox{and} \qquad \gamma_1=\sqrt{7}. $$ The elements of sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ and $\{\gamma_n\}_{n=1}^{\infty}$ satisfy the system of recurrence relations~(\ref{f2s-w1g}) and, selectively, recurrence relation~(\ref{f2s-w2g}). \end{lemma} The first twelve values of numbers $\alpha_{n}^{*}=\alpha_{n}/\sqrt{7}$, $\beta_{n}^{*}=\beta_{n}/\sqrt{7}$ and $\gamma_{n}^{*}=\gamma_{n}/\sqrt{7}$ are presented in Table~\ref{f2s-tab1}. \begin{remark} {\rm There exists a~simple relationships between numbers $\alpha_n$, $\beta_n$, $\gamma_n$, $n\in \mathbb{N}$, and $a_n$, $b_n$, $c_n$, $n\in \mathbb{N}$. We have $$ \alpha_{n} \equiv c_n,\qquad \beta_{n} \equiv {-}a_{n-1} - b_{n-1} ,\qquad \gamma_{n} \equiv a_{n-1}. $$ These relations are easily derived from the definitions of respective numbers, for example \begin{align*} \beta_{n} &= 2^{2(n-1)+1} \Big[ 2\, \sin (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2(n-1)}+{}\\ &\phantom{=}\hspace*{2ex} {}+ 2\, \sin (\tfrac{8\, \pi}{7})\, \cos (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2(n-1)}+%{}\\ %&\phantom{=}\hspace*{2ex} {}+ 2\, \sin (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2(n-1)} \Big]= \\ &= 2^{2(n-1)+1} \Big[ -\big(\sin (\tfrac{2\, \pi}{7}) + \sin (\tfrac{4\, \pi}{7})\big) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{2(n-1)}+{}\\ &\phantom{=}\hspace*{2ex} {} + \big({-}\sin (\tfrac{4\, \pi}{7}) - \sin (\tfrac{8\, \pi}{7})\big) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{2(n-1)}-%{}\\ %&\phantom{=}\hspace*{2ex} {} - \big( \sin (\tfrac{2\, \pi}{7}) + \sin (\tfrac{8\, \pi}{7}) \big) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{2(n-1)} \Big]= {}\\ &= - a_{n-1} - b_{n-1}. \\ \end{align*} } \end{remark} Let us also set \begin{align} f_{n} &= 2^{n+1}\, \Big[ \cos (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{n} + \cos (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{n} + \cos (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{n} \Big],\label{w2.11a}\\ g_{n} &= 2^{n+1}\, \Big[ \cos (\tfrac{8\, \pi}{7}) \big( \cos (\tfrac{4\, \pi}{7}) \big)^{n} + \cos (\tfrac{2\, \pi}{7}) \big( \cos (\tfrac{8\, \pi}{7}) \big)^{n} + \cos (\tfrac{4\, \pi}{7}) \big( \cos (\tfrac{2\, \pi}{7}) \big)^{n} \Big],\label{w2.11b}\\ h_{n} &= 2^{n+1}\, \Big[ \big( \cos (\tfrac{2\, \pi}{7}) \big)^{n+1} + \big( \cos (\tfrac{4\, \pi}{7}) \big)^{n+1} + \big( \cos (\tfrac{8\, \pi}{7}) \big)^{n+1} \Big],\label{w2.11} \end{align} for $n=0,1,2,\ldots$ \begin{lemma}\label{f2s-lem3} We have $$ f_0=g_0=h_0=-1,\qquad \mbox{and} \qquad h_1=5 $$ and \begin{equation}\label{f2s-w3g} \left\{ \begin{array}{ll} f_{n+1} = f_{n} + g_{n},& \quad n \geq 0,\\ g_{n+1} = f_{n} + h_{n},& \quad n \geq 0,\\ h_{n+1} = g_{n} + 2\, h_{n-1},& \quad n \geq 1.\\ \end{array} \right. \end{equation} The elements of sequences $\{f_n\}_{n=0}^{\infty}$, $\{g_n\}_{n=0}^{\infty}$ and $\{h_n\}_{n=0}^{\infty}$ satisfy the following recurrence relation (see formula~(\ref{w1-d})): \begin{equation}\label{ww3} x_{n+3}+x_{n+2}-2\, x_{n+1} - x_{n} =0,\qquad n=0,1,\ldots \end{equation} \end{lemma} %Proof \begin{proof} By~(\ref{f2s-w3g}) we obtain \begin{align} g_{n} & = f_{n+1}-f_{n},\label{ww1}\\ h_{n} & = g_{n+1}-f_{n} = f_{n+2}-f_{n+1}-f_{n},\label{ww2} \end{align} and, finally, the following identity $$ f_{n+3}-f_{n+2}-f_{n+1} = 2\, \big( f_{n+1} - f_{n} - f_{n-1} \big) + f_{n+1} - f_{n}, $$ i.e., \begin{equation}\label{ww4} f_{n+3}-f_{n+2}-4\, f_{n+1} +3\, f_{n} + 2\, f_{n-1} = 0. \end{equation} But we also have the following decomposition of respective characteristic polynomial $$ x^4-x^3-4\, x^2+3\, x+2 = (x-2)(x^3+x^2-2\, x-1), $$ which implies the following form of~(\ref{ww4}): \begin{equation}\label{ww5} f_{n+3}+f_{n+2}-2\, f_{n+1} - f_{n} = 2\, \big( f_{n+2} + f_{n+1} - 2\, f_{n} - f_{n-1} \big). \end{equation} Since $$ f_{3}+f_{2}-2\, f_{1} - f_{0} = 0 $$ so, we obtain the required identity~(\ref{ww3}) from~(\ref{ww5}), (\ref{ww1}) and~(\ref{ww2}). \end{proof} \bigskip The first twelve elements of the sequences $\{f_n\}_{n=0}^{\infty}$, $\{g_n\}_{n=0}^{\infty}$ and $\{h_n\}_{n=0}^{\infty}$ are presented in Table~\ref{f2s-tab1}. \begin{remark} {\rm We note, that \begin{equation} h_{n-1}=\mathcal{B}_{n} = \frac{1}{2} \Big( \big(\mathcal{A}_{n}\big)^{2} - \mathcal{A}_{2n} \Big),\qquad n=1,2,\ldots \end{equation} is an accelerator sequence for Catalan's constant (see~\cite{bradley} and \seqnum{A094648}~\cite{sloan}). } \end{remark} The next lemma contains a~sequence of eight identities and simultanously six newly defined auxiliary sequences of real numbers $\{\widetilde{A}_{n}\}$, $\{\widetilde{B}_{n}\}$, $\{\widetilde{C}_{n}\}$, $\{F_{n}\}$, $\{G_{n}\}$ and $\{H_{n}\}$. \begin{lemma}\label{lem2.6} The following identities hold % % \begin{multline}\label{lem2.6-w1} 4 \, \cos (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + 4 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} + {} \\ {} + 4 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} = {}\\ {} = f_{n}^{2} + g_{n}^{2} - h_{n}^{2} + h_{2n+1} -2\, h_{2n} - 4\, h_{2n-1} -f_{2n} := 2\, F_{n}, \end{multline} % % \begin{multline}\label{lem2.6-w2} 2 \, \cos (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} + 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\ {}+ 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} = {}\\ {} = F_{n} + h_{n}^{2} - f_{n}^{2} - h_{2n+1} + 2\, h_{2n} + 2\, h_{2n-1} := G_{n}, \end{multline} % % \begin{multline}\label{lem2.6-w3} 2 \, \cos (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\ {}+ 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} = {}\\ {} = f_{n}^{2} - 2\, ( h_{2n} + h_{2n-1} ) - F_{n} = h_{n}^{2} - h_{2n+1} - G_{n} := H_{n}, \end{multline} % % \begin{equation}\label{lem2.6-w4} g_{n}^{2} - 2\, h_{2n-1} - f_{2n} = (-1)^{n} \big( \mathcal{A}_{n} + \mathcal{A}_{n+1} - 7\, A_{n}\big) =G_{n}+F_{n}, \end{equation} % % \begin{equation}\label{lem2.6-w5} f_{n}^{2} - 2\, h_{2n} - 2\, h_{2n-1} = (-1)^{n} \big( \mathcal{A}_{n} - \mathcal{A}_{n-2} \big) =H_{n}+F_{n}, \end{equation} % % \begin{multline}\label{lem2.6-w6} 2 \, \sin (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} + 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + {} \\ {}+ 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} = {}\\ {} = a_{2n} -a_{n}\, h_{2n-1} + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} ) := \widetilde{A}_{n}, \end{multline} % % \begin{multline}\label{lem2.6-w7} 2 \, \sin (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} + {} \\ {}+ 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} = {}\\ {} = b_{2n}-b_{n}\, h_{2n-1} + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} ) := \widetilde{B}_{n}, \end{multline} % % \begin{multline}\label{lem2.6-w8} 2 \, \sin (\tfrac{2\, \pi}{7})\, \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + {} \\ {}+ 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} = {}\\ {} = c_{2n}-c_{n}\, h_{2n-1} + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} ) := \widetilde{C}_{n}, \end{multline} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Now we are ready to present the final result of this section. All recurrent sequences defined in this section are applied below to the description of the coefficients of certain polynomials. \begin{theorem}\label{f2s-tw1} The following decompositions of polynomials hold: \begin{multline}\label{f2s-tw1-w-a} \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \left\{ \begin{array}{ll} \mathbb{X}^3 - a_k\, \mathbb{X}^2 + 7\, B_{2k}\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\ \mathbb{X}^3 - \alpha_{k-1}\, \mathbb{X}^2 - 7\, B_{2k-1}\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k-1,\\ \end{array} \right. \end{multline} % % \begin{multline}\label{f2s-tw1-w-b} \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \left\{ \begin{array}{ll} \mathbb{X}^3 - b_k\, \mathbb{X}^2 + 7\,( C_{2k} - B_{2k} )\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\ \mathbb{X}^3 - \beta_{k-1}\, \mathbb{X}^2 + 7\, (B_{2k-1} - C_{2k-1})\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k-1,\\ \end{array} \right. \end{multline} % % \begin{multline}\label{f2s-tw1-w-c} \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \left\{ \begin{array}{ll} \mathbb{X}^3 - c_k\, \mathbb{X}^2 - 7\, C_{2k} \, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\ \mathbb{X}^3 - \gamma_{k-1}\, \mathbb{X}^2 + 7\, C_{2k-1}\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k-1,\\ \end{array} \right. \end{multline} % % \begin{multline}\label{f2s-tw1-w-d} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - f_n\, \mathbb{X}^2 + (-1)^{n} \, ( 7\,{A}_{n} - 3\,\mathcal{A}_{n} ) \, \mathbb{X} - 1, \end{multline} % % \begin{multline}\label{f2s-tw1-w-e} \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - g_n\, \mathbb{X}^2 + (-1)^{n} \, ( \mathcal{A}_{n} + \mathcal{A}_{n+1} - 7\, A_{n} ) \, \mathbb{X} - 1, \end{multline} % % \begin{multline}\label{f2s-tw1-w-f} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)\times {} \\ {} \times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - {F}_n\, \mathbb{X}^2 + ( f_{n} + h_{n} ) \, \mathbb{X} - 1, \end{multline} % % \begin{multline}\label{f2s-tw1-w-g} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)\times {} \\ {} \times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - {G}_n\, \mathbb{X}^2 + ( f_{n} + g_{n} ) \, \mathbb{X} - 1, \end{multline} % % \begin{multline}\label{f2s-tw1-w-h} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {} \\ {} \times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - {H}_n\, \mathbb{X}^2 + ( g_{n} + h_{n} ) \, \mathbb{X} - 1. \end{multline} \end{theorem} \begin{proof} Ad~(\ref{f2s-tw1-w-d}) We have \begin{equation*} 4 \, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) = 2 \, \cos (\tfrac{2\, \pi}{7}) + 2 \, \cos (\tfrac{4\, \pi}{7}) \end{equation*} hence, by~(\ref{w1-d}) we obtain \begin{multline*} \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n+1} = \big( 2 \, \cos (\tfrac{2\, \pi}{7}) + 2 \, \cos (\tfrac{4\, \pi}{7}) \big) \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} = {} \\ {}=\Big[ \big( 2 \, \cos (\tfrac{2\, \pi}{7}) + 2 \, \cos (\tfrac{4\, \pi}{7}) + 2 \, \cos (\tfrac{8\, \pi}{7}) \big) - \big( 2 \, \cos (\tfrac{2\, \pi}{7}) + 2 \, \cos (\tfrac{8\, \pi}{7}) \big) + 2 \, \cos (\tfrac{2\, \pi}{7}) \Big]\times {} \\ {}\times \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} %= {} \\ {}= - \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} - 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\ + \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n-1}. \end{multline*} Treating, in a similar way, the following products: \begin{equation*} \big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{4\, \pi}{7}) \big)^{n+1} \qquad \mbox{ and } \qquad \big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n+1}, \end{equation*} and adding all three received decompositions, by~(\ref{wzor-gw1new}), we obtain \begin{multline*} (-1)^{n}\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n} - \mathcal{A}_{n-1}\big) %= {}\\ = 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {}\\ {} + 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{4\, \pi}{7}) \big)^{n} %+ {}\\ + 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n}. \end{multline*} But, by~\cite{wsw1}, we have $$ \mathcal{A}_{n+3} - 2\, \mathcal{A}_{n+2} - \mathcal{A}_{n+1} + \mathcal{A}_{n} \equiv 0, $$ hence $$ \mathcal{A}_{n+1} - \mathcal{A}_{n} - \mathcal{A}_{n-1} = \mathcal{A}_{n} - \mathcal{A}_{n-2}, $$ which implies decomposition~(\ref{f2s-tw1-w-d}). \end{proof} \bigskip \begin{remark} {\rm It should also be noted that the following interesting identity holds: \begin{equation} 4\, \mathcal{A}_{n} - \mathcal{A}_{n-2} = 7\, A_{n}. \end{equation} } \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The second group of special cases of~(\ref{choinka})} Let us set \begin{align} u_{n} &= 2\, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.1}\\ v_{n} &= 2\, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.2}\\ w_{n} &= 2\, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.3}\\ x_{n} &= 2\, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.4}\\ y_{n} &= 2\, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.5}\\ z_{n} &= \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n+1} + \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n+1} + \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n+1},\label{w4.6} \end{align} for $n\in \mathbb{N}$ and $u_0=v_0=w_0=-1$ and $z_0=y_0=z_0=\sqrt{7}$, $z_1=7$ (see~Table~\ref{f2s-tab3}, and for sequence $\{-w_{2n+1}/\sqrt{7}\}$ see \seqnum{A115146}~\cite{sloan}, and see \seqnum{A079309}~\cite{sloan} for sequence $\{z_{2n}/\sqrt{7}\}$). Then, as may by verified without difficulty, the following recurrence relations hold \begin{equation} \left\{ \begin{array}{l} u_{n+1} = x_{n},\\ v_{n+1} = -y_{n}-z_{n} = x_{n} - \sqrt{7} \, z_{n-1},\\ w_{n+1} = y_{n}-x_{n},\\ x_{n+1} = u_{n} - w_{n},\\ y_{n+1} = w_{n} - v_{n},\\ z_{n+1} = 2\, z_{n-1} - v_{n}, \end{array} \right. \end{equation} for every $n\in \mathbb{N}$. Hence, we easily obtain \begin{equation} \left\{ \begin{array}{l} x_{n+2} = x_{n} - w_{n+1} = 2\, x_{n} - y_{n},\\ y_{n+2} = y_{n} - v_{n+1} - x_{n} = 2\, y_{n} - x_{n} + z_{n},\\ z_{n+2} = y_{n} + 3\, z_{n}, \end{array} \right. \end{equation} for every $n\in \mathbb{N}$ and finally the recurrence relation (see also equation~(\ref{w1-b})): \begin{equation}\label{w3n.9} z_{n+6} - 7\, z_{n+4} + 14\, z_{n+2} - 7\, z_{n} = 0, \end{equation} which also satisfies the remaining sequences discussed in this section: $\{x_n\}$, $\{y_n\}$, $\{u_n\}$, $\{v_n\}$ and $\{w_n\}$. \begin{remark} {\rm The characteristic polynomial of equation~(\ref{w3n.9}) (after rescaling) has the form of~(\ref{w1-b}) and was recognized by Johannes Kepler (1571--1630). The roots of this polynomial are equal to $|A_1A_2|^2$, $|A_1A_3|^2$ and $|A_1A_4|^2$, where $A_1A_2\ldots A_7$ is a~regular convex heptagon inscribed in the unit circle~\cite{GrzymkowskiWitula}. } \end{remark} \begin{theorem}\label{f2s-tw2} The following decompositions of polynomials hold: \begin{multline}\label{f2s-tw2-w-a} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - u_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( u_{n}^{2} - v_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n}, \end{multline} % % \begin{multline}\label{f2s-tw2-w-b} \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - v_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( v_{n}^{2} - w_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n}, \end{multline} % % \begin{multline}\label{f2s-tw2-w-c} \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - w_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( w_{n}^{2} - u_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n}, \end{multline} % % \begin{multline}\label{f2s-tw2-w-d} \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - x_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( x_{n}^{2} + w_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n+1}, \end{multline} % % \begin{multline}\label{f2s-tw2-w-e} \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - y_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( y_{n}^{2} + u_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n+1}, \end{multline} % % \begin{multline}\label{f2s-tw2-w-f} \big( \mathbb{X} - \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n+1} \big) \big( \mathbb{X} - \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n+1} \big) \big( \mathbb{X} - \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n+1} \big) = {}\\ {} = \mathbb{X}^3 - z_n\, \mathbb{X}^2 +\tfrac{1}{2} \big( z_{n}^{2} - z_{2n+1} \big)\, \mathbb{X} - (-\sqrt{7}\, )^{n+1}. \end{multline} % % \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Additionally, primarily to describe the generating functions of the sequences $\{u_n\}$, $\{v_n\}$, $\{w_n\}$, $\{x_n\}$, $\{y_n\}$ and $\{z_n\}$ (see Section~\ref{roz-fun-two}) six new recurrence sequences were introduced, which, from a certain point of view, are conjugated with sequences $\{u_n\}$, \ldots, $\{z_n\}$. Let us set \begin{align} u_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + {}\nonumber \\ &\hspace*{10mm} {}+ 2\, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n},\label{w4.16a}\\ v_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + {}\nonumber \\ &\hspace*{10mm} {}+ 2\, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.17}\\ w_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + 2\, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +{}\nonumber \\ &\hspace*{10mm} {}+ 2\, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.18}\\ x_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +{}\nonumber \\ &\hspace*{10mm} {}+ 2\, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.19}\\ y_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +{}\nonumber \\ &\hspace*{10mm} {}+ 2\, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.20}\\ z_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + 2\, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +{}\nonumber \\ &\hspace*{10mm} {}+ 2\, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n},\label{w4.21} \end{align} for $n\in \mathbb{N}$ and $u_{0}^{*}=v_{0}^{*}=w_{0}^{*}=-1$ and $x_{0}^{*}=y_{0}^{*}=z_{0}^{*}=\sqrt{7}$ (see~Table~\ref{f2s-tab2}). An easy computation shows that the following recurrence relations hold \begin{equation} \left\{ \begin{array}{l} u_{n+1}^{*} = u_{n}^{*}+w_{n}^{*}-v_{n}^{*}- z_{n-1}^{2} + z_{2n-1},\\ v_{n+1}^{*} = z_{n-1}^{2} - z_{2n-1} - u_{n}^{*},\\ w_{n+1}^{*} = u_{n}^{*}-w_{n}^{*}, \end{array} \right. \end{equation} for every $n \geq 1$. Hence, we obtain \begin{align} &u_{n+1}^{*}+v_{n+1}^{*} = w_{n}^{*} - v_{n}^{*},\\ &u_{n}^{*} = w_{n+1}^{*} + w_{n}^{*},\\ &w_{n+2}^{*} + w_{n+1}^{*} - w_{n}^{*} = -v_{n+1}^{*} - v_{n}^{*},\\ &w_{n+3}^{*} -3\, w_{n+1}^{*} - w_{n}^{*} = z_{2n+1} + z_{2n-1} -z_{n}^{2} - z_{n-1}^{2}. \end{align} for every $n \geq 1$. Also the following recurrence relations hold \begin{equation} \left\{ \begin{array}{l} x_{n+1}^{*} = 2\, y_{n}^{*} - z_{n}^{*},\\ y_{n+1}^{*} = 2\, x_{n}^{*}+y_{n}^{*}-z_{n}^{*},\\ z_{n+1}^{*} = -x_{n}^{*}-y_{n}^{*}-z_{n}^{*}, \end{array} \right. \end{equation} for every $n \in \mathbb{N}$, which implies the identities \begin{align} &y_{n+1}^{*} = -x_{n+2}^{*} +2\, x_{n+1}^{*} + 7\, x_{n}^{*},\\ &x_{n+2}^{*} + 7\, x_{n}^{*} + 7\, x_{n-1}^{*} = 0 \end{align} for every $n \geq 1$. \begin{theorem}\label{f2s-tw2a} The following decompositions of polynomials hold: \begin{multline}\label{w4.30} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - u_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( u_{n} + v_{n} \big)\, \mathbb{X} - 7^{n}, \end{multline} % \begin{multline}\label{w4.31} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - v_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( v_{n} + w_{n} \big)\, \mathbb{X} - 7^{n}, \end{multline} % \begin{multline}\label{w4.32} \big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - w_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( u_{n} + w_{n} \big)\, \mathbb{X} - 7^{n}, \end{multline} % \begin{multline}\label{w4.33} \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - x_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( w_{n} - v_{n} \big)\, \mathbb{X} - 7^{n}, \end{multline} % \begin{multline}\label{w4.34} \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - y_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( u_{n} - w_{n} \big)\, \mathbb{X} - 7^{n}, \end{multline} % % \begin{multline}\label{w4.35} \big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7}) \big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\ {}\times \big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7}) \big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\ {} = \mathbb{X}^3 - z_{n}^{*}\, \mathbb{X}^2 +(-\sqrt{7}\, )^{n}\,\big( v_{n} - w_{n} \big)\, \mathbb{X} - 7^{n}. \end{multline} % \end{theorem} \begin{lemma} The following two groups of identities hold: \begin{equation} \begin{split} u_{n}^{2} & = 2\, z_{2n-1} + v_{2n} + v_{n}^{*} + u_{n}^{*},\\ v_{n}^{2} & = 2\, z_{2n-1} + w_{2n} + v_{n}^{*} + w_{n}^{*},\\ w_{n}^{2} & = 2\, z_{2n-1} + u_{2n} + u_{n}^{*} + w_{n}^{*},\\ x_{n}^{2} & = 2\, z_{2n-1} - w_{2n} + w_{n}^{*} - v_{n}^{*},\\ y_{n}^{2} & = 2\, z_{2n-1} - u_{2n} + u_{n}^{*} - w_{n}^{*},\\ z_{n}^{2} & = 2\, z_{2n-1} + 2\, v_{n}^{'} -2 \, u_{n}^{*}, \end{split} \end{equation} and \begin{equation} \begin{split} \big(u_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + v_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} + v_{n} \big),\\ \big(v_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + w_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( v_{n} + w_{n} \big),\\ \big(w_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + u_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} + w_{n} \big),\\ \big(x_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - w_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( w_{n} - v_{n} \big),\\ \big(y_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - u_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} - w_{n} \big),\\ \big(z_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - v_{n}^{*} + 2\, \big({-}\sqrt{7}\,\big)^{n} \big( v_{n} - u_{n} \big). \end{split} \end{equation} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 5 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Some Ramanujan-type formulas} Let $p,q,r\in \mathbb{R}$. Shevelev~\cite{Shevelev} (see also~\cite{GrzymkowskiWitula}) proved that if $z_1$, $z_2$, $z_3$ are roots of the polynomial $$ w(z)=z^3+p\, z^2+q\, z+r $$ and $z_1$, $z_2$, $z_3$ are all reals and at least the following condition holds \begin{equation}\label{w-srf1} p\, \sqrt[3]{r} + 3\, \big( \sqrt[3]{r} \big)^2 + q =0, \end{equation} then the following formula holds \begin{multline}\label{w-srf2} \sqrt[3]{z_1} + \sqrt[3]{z_2} + \sqrt[3]{z_3} = \sqrt[3]{-p-6\,\sqrt[3]{r} + \sqrt[3]{(p+6\, \sqrt[3]{r})^3 - (p-3\, \sqrt[3]{r})^3} }= {}\\ {}= \sqrt[3]{-p-6\,\sqrt[3]{r} + 3\, \sqrt[3]{9\, r + 3\, p\, (\sqrt[3]{r})^2 + p^2\, \sqrt[3]{r}} } \end{multline} (all radicals are determined to be real). It was verified that only following polynomials (among all discussed in this paper) satisfy the condition~(\ref{w-srf1}); simultanously below the respective form of identity~(\ref{w-srf2}) is presented: \bigskip \noindent polynomial (\ref{w1-d}) and polynomial~(\ref{wzor-gw1new}) for $n=1$ (it is the classical Ramanujan formula): \begin{equation}\label{w-fib1} \sqrt[3]{2\,\cos ( \tfrac{2\, \pi}{7})} + \sqrt[3]{2\,\cos ( \tfrac{4\, \pi}{7})} + \sqrt[3]{2\,\cos ( \tfrac{8\, \pi}{7})} = \sqrt[3]{5-3\, \sqrt[3]{7}}; \end{equation} \noindent polynomial (\ref{f2s-tw1-w-e}) for $n=4$: \begin{equation}\label{w-fib2} \cos ( \tfrac{2\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{8\, \pi}{7})} + \cos ( \tfrac{4\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{2\, \pi}{7})} + \cos ( \tfrac{8\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{4\, \pi}{7})} = \frac{1}{2}\, \sqrt[3]{18\, (2-\sqrt[3]{7})}; \end{equation} \noindent polynomial (\ref{f2s-tw1-w-e}) for $n=3$: \begin{multline}\label{w-fib3} 2\,\cos ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + 2\,\cos ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} + 2\,\cos ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} ={}\\ {}= \sqrt[3]{-2-3\,\sqrt[3]{49}}; \end{multline} \noindent polynomial (\ref{f2s-tw1-w-f}) for $n=2,3$, respectively: \begin{equation}\label{w-fib4} \sqrt[3]{\frac{\cos ( \tfrac{2\, \pi}{7})}{1+\cos ( \tfrac{8\, \pi}{7})}} + \sqrt[3]{\frac{\cos ( \tfrac{4\, \pi}{7})}{1+\cos ( \tfrac{2\, \pi}{7})}} + \sqrt[3]{\frac{\cos ( \tfrac{8\, \pi}{7})}{1+\cos ( \tfrac{4\, \pi}{7})}} = \sqrt[3]{11-3\,\sqrt[3]{49}}, \end{equation} \begin{equation}\label{w-fib5} \sec ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} + \sec ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + \sec ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} = -2\,\sqrt[3]{9\,(1+\sqrt[3]{7})}; \end{equation} \noindent polynomial~(\ref{f2s-tw1-w-d}) for $n=1$, polynomial (\ref{f2s-tw1-w-e}) for $n=1$ and polynomial (\ref{f2s-tw1-w-h}) for $n=2$: \begin{equation}\label{w-fib6} \sqrt[3]{\sec ( \tfrac{2\, \pi}{7})} + \sqrt[3]{\sec ( \tfrac{4\, \pi}{7})} + \sqrt[3]{\sec ( \tfrac{8\, \pi}{7})} = \sqrt[3]{8-6\,\sqrt[3]{7}}; \end{equation} \noindent polynomial (\ref{f2s-tw2-w-b}) for $n=6$: \begin{multline}\label{w-fib7} \sin^2 ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + \sin^2 ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} + \sin^2 ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} ={}\\ {}= -\frac{1}{4}\, \sqrt[3]{63\, (1+\sqrt[3]{7})}; \end{multline} \noindent polynomial (\ref{f2s-tw2-w-d}) for $n=2$: \begin{equation}\label{w-fib8} \sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}} + \sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}} + \sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}} = \sqrt[3]{5-3\,\sqrt[3]{7}}; \end{equation} \noindent polynomial (\ref{f2s-tw2-w-e}) for $n=2$: \begin{equation}\label{w-fib9} \sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}} + \sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}} + \sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}} = \sqrt[3]{4-3\,\sqrt[3]{7}}; \end{equation} \noindent polynomial (\ref{f2s-tw2-w-e}) for $n=5$: \begin{equation} \sin^2 (\tfrac{2\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}} + \sin^2 (\tfrac{4\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}} + \sin^2 (\tfrac{8\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}} = \frac{1}{4}\, \sqrt[3]{77-21\,\sqrt[3]{49}}; \end{equation} \noindent polynomial (\ref{w4.30}) for $n=3$: \begin{multline} \sqrt[6]{7}\, \Big( \csc(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} + \csc(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + \csc(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} \Big) ={}\\ {}= 2\, \sqrt[3]{2+3\,\sqrt[3]{49}}; \end{multline} \noindent polynomial (\ref{w4.32}) for $n=3$: \begin{multline} \sqrt[6]{7}\, \Big( \csc(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} + \csc(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + \csc(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} \Big) ={}\\ {}= 2\, \sqrt[3]{3\,\sqrt[3]{7}-5}; \end{multline} \noindent polynomial (\ref{w4.32}) for $n=6$: \begin{multline} \csc^2(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} + \csc^2(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} + \csc^2(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} ={}\\ {}= -4\, \sqrt[3]{\frac{2+3\,\sqrt[3]{49}}{7}}. \end{multline} \begin{remark} {\rm Shevelev~\cite{Shevelev} presented the identities~(\ref{w-fib8}) and~(\ref{w-fib9}) in the following alternative form: \begin{equation} \sqrt[3]{2+\sec ( \tfrac{2\, \pi}{7})} + \sqrt[3]{2+\sec ( \tfrac{4\, \pi}{7})} + \sqrt[3]{2+\sec ( \tfrac{8\, \pi}{7})} = \sqrt[3]{6\,\sqrt[3]{7}-10} \end{equation} and \begin{multline} \sqrt[3]{1+2\,\cos ( \tfrac{2\, \pi}{7})} + \sqrt[3]{1+2\,\cos ( \tfrac{4\, \pi}{7})} + \sqrt[3]{1+2\,\cos ( \tfrac{8\, \pi}{7})} ={}\\ {}= \sqrt[3]{\frac{2\cos ( \tfrac{2\, \pi}{7})}{2\cos ( \tfrac{2\, \pi}{7})+1}} + \sqrt[3]{\frac{2\cos ( \tfrac{4\, \pi}{7})}{2\cos ( \tfrac{4\, \pi}{7})+1}} + \sqrt[3]{\frac{2\cos ( \tfrac{8\, \pi}{7})}{2\cos ( \tfrac{8\, \pi}{7})+1}} = \sqrt[3]{3\,\sqrt[3]{7}-4}, \end{multline} respectively. } \end{remark} \begin{remark} {\rm By~(\ref{w-fib7}),~(\ref{w-fib1}) and by applied the formula $2\, \sin^2 \alpha = 1-\cos (2\alpha)$ we obtain \begin{multline} 2\, \cos \big( \tfrac{2\, \pi}{7} \big)\, \sqrt[3]{2\, \cos \big( \tfrac{2\, \pi}{7} \big)\,} + 2\, \cos \big( \tfrac{4\, \pi}{7} \big)\, \sqrt[3]{2\, \cos \big( \tfrac{4\, \pi}{7} \big)\,} + 2\, \cos \big( \tfrac{8\, \pi}{7} \big)\, \sqrt[3]{2\, \cos \big( \tfrac{8\, \pi}{7} \big)\,} = {} \\ {}= \sqrt[3]{63\, \big(1+\sqrt[3]{7}\big)\,} -2\,\sqrt[3]{5 - 3\, \sqrt[3]{7}\,}. \end{multline} Moreover, from~(\ref{w-fib1}),~(\ref{w-fib6}) and the equality $8\, \cos (\frac{2\pi}{7}) \, \cos (\frac{4\pi}{7}) \, \cos (\frac{8\pi}{7}) = 1$ we get \begin{multline} \sqrt[3]{ \Big(2\, \cos \big( \tfrac{2\, \pi}{7} \big)\Big)^2\,} + \sqrt[3]{\Big(2\, \cos \big( \tfrac{4\, \pi}{7} \big)\Big)^2\,} + \sqrt[3]{\Big(2\, \cos \big( \tfrac{8\, \pi}{7} \big)\Big)^2\,} = {} \\ {}= \sqrt[3]{\Big( 5 - 3\, \sqrt[3]{7} \Big)^2\,} -2\,\sqrt[3]{4 - 3\, \sqrt[3]{7}\,}. \end{multline} } \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 6 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The sine-Fibonacci numbers of order $7$} The quasi-Fibonacci numbers of order $7$ introduced by Witu{\l}a et al.\ in~\cite{wsw1} and further developed in Section~\ref{roz1} constitute the simplest, only one-parameter type of the so-called cosine-Fibonacci numbers of order $7$. The name ``cosine-Fibonacci numbers'' is derived from the form of the decomposition of formulas: $(1+\delta\, (\xi+\xi^6))^{n}=(1+2\, \delta\, \cos(\tfrac{2\pi}{7}))^{n}$,~\ldots In this Section we shall introduce and analyze a sine variety of these numbers, created in the course of decomposing formulas: $(1+\delta\, (\xi-\xi^6))^{n}=(1+2\, i\, \delta\, \sin(\tfrac{2\pi}{7}))^{n}$,~\ldots \begin{theorem} Let $\xi, \delta \in \mathbb{C}$, $\xi^7=1$ and $\xi \neq 1$. Then, for every $n\in \mathbb{N}$, there exist polynomials $p_n, r_n, s_n, k_n, l_n \in \mathbb{Z}[\delta]$, called here the sine-Fibonacci numbers of order $7$, so that \begin{align} (1+\delta\, (\xi^{\phantom{2}}-\xi^6))^n &= p_n(\delta)+r_n(\delta)\, (\xi^{\phantom{2}}-\xi^6)+s_n(\delta)\, (\xi^2-\xi^5)+{}\nonumber\\ &\phantom{=}\hspace*{2ex}{} +k_n(\delta)\, (\xi^{\phantom{2}}+\xi^6)+l_n(\delta)\, (\xi^2+\xi^5),\\ (1+\delta\, (\xi^2 -\xi^5))^n&= p_n(\delta)+r_n(\delta)\, (\xi^{2}-\xi^5)+s_n(\delta)\, (\xi^4-\xi^3)+{}\nonumber\\ &\phantom{=}\hspace*{2ex}{} +k_n(\delta)\, (\xi^{2}+\xi^5)+l_n(\delta)\, (\xi^4+\xi^3),\\ (1+\delta\, (\xi^4 -\xi^3))^n&= p_n(\delta)+r_n(\delta)\, (\xi^{4}-\xi^3)+s_n(\delta)\, (\xi^{\phantom{2}}-\xi^6)+{}\nonumber\\ &\phantom{=}\hspace*{2ex}{} +k_n(\delta)\, (\xi^{4}+\xi^3)+l_n(\delta)\, (\xi^{\phantom{2}}+\xi^6). \end{align} We have $p_1(\delta)=1$, $r_1(\delta)=\delta$ and $s_1(\delta)=k_1(\delta)=l_1(\delta)=0$ (see Table~\ref{f2s-tab4}, where the initial values of these polynomial are presented). These polynomials are connected by recurrence relations \begin{equation}\label{f2s-r6-w1g} \left\{ \begin{array}{l} p_{n+1}(\delta) = p_{n}(\delta) - \delta\, s_{n}(\delta) - 2\, \delta\, r_{n}(\delta) - i\, \sqrt{7}\, \delta l_{n}(\delta),\\ r_{n+1}(\delta) = r_{n}(\delta) + \delta\, p_{n}(\delta),\\ s_{n+1}(\delta) = s_{n}(\delta) + \delta\, k_{n}(\delta) + \delta\, l_{n}(\delta),\\ k_{n+1}(\delta) = k_{n}(\delta) - 2\, \delta\, s_{n}(\delta),\\ l_{n+1}(\delta) = l_{n}(\delta) + \delta\, r_{n}(\delta) - \delta\, s_{n}(\delta),\\ \end{array} \right. \end{equation} for $n\in \mathbb{N}$. Hence, the following relationships can be deduced: \begin{align} \delta\, p_{n}(\delta) & = r_{n+1}(\delta) - r_{n}(\delta),\label{w5.4.a}\\ 2\,\delta\, s_{n}(\delta) & = k_{n+1}(\delta) - k_{n}(\delta),\label{w5.4.b}\\ 2\,\delta^2\, l_{n}(\delta) & = -k_{n+2}(\delta) +2 \, k_{n+1}(\delta) -(1+2\, \delta^2)\, k_{n}(\delta),\label{w5.4.c}\\ 2\,\delta^3\, r_{n}(\delta) & = -k_{n+3}(\delta) + 3 \, k_{n+2}(\delta) -3\,(1+\delta^2)\, k_{n+1}(\delta) +(1+3\, \delta^2)\, k_{n}(\delta)\label{w5.4.d} \end{align} and, at last, the main identity \begin{multline}\label{f2s-r6-wdelta} k_{n+5}(\delta) - 5\, k_{n+4} + (10+5\, \delta^2)\, k_{n+3}(\delta)+ (i\, \sqrt{7} \, \delta^3 -15\, \delta^2 -10)\, k_{n+2}(\delta)+\\ +(7\, \delta^4 - i\, 2\, \sqrt{7} \, \delta^3 + 15\, \delta^2 + 5)\, k_{n+1}(\delta)+\\ +(i\, 2\, \sqrt{7} \, \delta^5 - 7\, \delta^4 + i\, \sqrt{7} \, \delta^3 - 5\, \delta^2 - 1)\, k_{n}(\delta) =0. \end{multline} This identity satisfies, also by~(\ref{w5.4.a})--(\ref{w5.4.d}), the remaining polynomials: $p_{n}(\delta)$, $r_{n}(\delta)$, $s_{n}(\delta)$, and $l_{n}(\delta)$, $n\in \mathbb{N}$. The characteristic polynomial corresponding to identity~(\ref{f2s-r6-wdelta}) has the following decomposition: \begin{multline}\label{wz6.6} \mathbb{X}^5- 5\, \mathbb{X}^4 + (10+5\, \delta^2)\, \mathbb{X}^3+ (i\, \sqrt{7} \, \delta^3 -15\, \delta^2 -10)\, \mathbb{X}^2 +(7\, \delta^4 - i\, 2\, \sqrt{7} \, \delta^3 + 15\, \delta^2 + 5)\, \mathbb{X} + {}\\ {}+ i\, 2\, \sqrt{7} \, \delta^5 - 7\, \delta^4 + i\, \sqrt{7} \, \delta^3 - 5\, \delta^2 - 1= %{}\\ %{} = (\mathbb{X}-1-\delta\, (\xi-\xi^6)) (\mathbb{X}-1-\delta\, (\xi^2-\xi^5)) \times{}\\ {}\times (\mathbb{X}-1-\delta\, (\xi^4-\xi^3)) (\mathbb{X}-1+\delta\, (\xi+\xi^2+\xi^4)) (\mathbb{X}-1+\delta\, (\xi^3+\xi^5+\xi^6)) = {} \\ {} = \big( \mathbb{X}^3 + (-3 - i\, \sqrt{7}\, \delta) \, \mathbb{X}^2 + (3 + i\, 2\, \sqrt{7}\, \delta)\, \mathbb{X} -i\, \sqrt{7}\, \delta^3 - i\, \sqrt{7}\, \delta -1 \big) \times {}\\ {}\times \big( \mathbb{X}^2 + (-2 + i\, \sqrt{7}\, \delta) \, \mathbb{X} + 1 - 2\, \delta^2 - i\, \sqrt{7}\, \delta \big). \end{multline} Hence, we obtain, for example, the following explicit form of $k_{n}(\delta)$: \begin{multline} k_{n}(\delta) = a\,(1+\delta\, (\xi-\xi^6))^{n} + b\,(1+\delta\, (\xi^2-\xi^5))^{n} + {}\\ {} + c\,(1+\delta\, (\xi^4-\xi^3))^{n} + d\,(1-\delta\, (\xi+\xi^2+\xi^4))^{n} + e\,(1+\delta\, (\xi^3+\xi^5+\xi^6))^{n}, \end{multline} where \begin{align*} a & = -2 \big( 5+(\xi^2+\xi^5) +7\, (\xi^3+\xi^4) \big)^{-1} =\tfrac{2}{301} \big( 46\, (\xi+\xi^6)+5\,(\xi^3+\xi^4) -11\big),\\ b & = \phantom{-}2 \big( 2+7\, (\xi^2+\xi^5) +6\, (\xi^3+\xi^4) \big)^{-1} =\tfrac{2}{187} \big(19\, (\xi+\xi^6)+46\,(\xi^2+\xi^5) -5\big),\\ c & = -2 \big( 4+6\, (\xi^2+\xi^5) - (\xi^3+\xi^4) \big)^{-1} =\tfrac{2}{301} \big(5\, (\xi^2+\xi^5)+46\,(\xi^3+\xi^4) -11\big),\\ d & = -2 \big( {-5}+2\, (\xi+\xi^2+\xi^4) \big)^{-1}=\tfrac{2}{43} \big( 5-2\, (\xi^3+\xi^5+\xi^6) \big) ,\\ e & = \phantom{-}2 \big( 7+2\, (\xi+\xi^2+\xi^4) \big)^{-1}=\tfrac{2}{43} \big( 7+2\, (\xi^3+\xi^5+\xi^6) \big). \end{align*} \end{theorem} \begin{corollary} The following identity holds: \begin{multline}\label{wz6.8} \Omega_{n} (\delta) := (1+\delta\, (\xi-\xi^6))^{n} + (1+\delta\, (\xi^2-\xi^5))^{n} + (1+\delta\, (\xi^4-\xi^3))^{n} = {}\\ {} = 3\, p_{n}(\delta) - k_{n}(\delta) - l_{n}(\delta) + i\, \sqrt{7} \big( r_{n}(\delta) + s_{n}(\delta) \big). \end{multline} \end{corollary} \begin{definition} To simplify the notation, we shall write $\Omega_n$ instead of $\Omega_{n}(1)$. \end{definition} \begin{theorem} The following decompositions of the polynomials hold: \begin{multline} q_{n}(\mathbb{X}; \delta):=\\ =\big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{2\, \pi}{7}))^{n}\big)\big) \big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{4\, \pi}{7}))^{n}\big)\big) \big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{8\, \pi}{7}))^{n}\big)\big) = {}\\ {} = \mathbb{X}^3 - \Omega_{n} (\delta)\, \mathbb{X}^2 +\tfrac{1}{2} \big( ( \Omega_{n} (\delta) )^{2} - \Omega_{2n} (\delta) \big)\, \mathbb{X} - \big(1+i\, \sqrt{7} \, \delta + i\, \sqrt{7} \, \delta^3 \big)^{n},\label{w5n.9} \end{multline} % \begin{multline}\label{w6.14} \big(x- \big( \cot (\tfrac{2\, \pi}{7}) \big)^{n}\big) \big(x- \big( \cot (\tfrac{4\, \pi}{7}) \big)^{n}\big) \big(x- \big( \cot (\tfrac{8\, \pi}{7}) \big)^{n}\big) = {}\\ {} = x^3 - \big( \tfrac{3}{\sqrt{7}} \big)^{n} \, \mathcal{A}_{n}(\tfrac{2}{3})\, x^2 + \Omega_{n} (\tfrac{2\, i}{\sqrt{7}})\, x +\frac{(-1)^{n-1}}{\big(\sqrt{7}\, \big)^{n}}. \end{multline} % Moreover, the following identity holds: \begin{multline} \Omega_{n}^{2}(\delta) = \Omega_{2n}(\delta) + 2\, \big( 3\, p_{n}^{2}(\delta) - 2\, l_{n}^{2}(\delta) - 2\, k_{n}^{2}(\delta) -2\, p_{n}(\delta)\, k_{n}(\delta) - 2\, p_{n}(\delta)\, l_{n}(\delta) - 7 \, r_{n}(\delta)\, s_{n}(\delta) + {} \\ {} + 3\, k_{n}(\delta)\, l_{n}(\delta) + i\, 2\, \sqrt{7}\, ( p_{n}(\delta)\, r_{n}(\delta) + p_{n}(\delta)\, s_{n}(\delta) - r_{n}(\delta)\, k_{n}(\delta) - s_{n}(\delta)\, l_{n}(\delta) ) + {} \\ {} + i\, \sqrt{7} ( r_{n}(\delta)\, l_{n}(\delta) - k_{n}(\delta)\, s_{n}(\delta) ) \big) = {} \\ {} = \Omega_{2n}(\delta) + 4\, p_{n}(\delta)\, \Omega_{n}(\delta) + 2\, \big( -3\, p_{n}^{2}(\delta) - 2\, l_{n}^{2} (\delta)- 2\, k_{n}^{2}(\delta) - 7\, r_{n}(\delta)\, s_{n}(\delta) + 3\, k_{n}(\delta)\, l_{n}(\delta) + {} \\ {} + i\, \sqrt{7} ( r_{n}(\delta)\, l_{n}(\delta) - 2\, r_{n}(\delta)\, k_{n}(\delta) - k_{n}(\delta)\, s_{n}(\delta) - 2\, s_{n}(\delta)\, l_{n}(\delta)) \big). \end{multline} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 7 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Generating functions}\label{roz-fun-two} The generating functions of almost all sequences discussed and defined in this paper are presented in this section. By~(\ref{w-and}) and~(\ref{w1.10}) we obtain (for~$k\in \mathbb{N}$): \begin{multline} \sum_{n=0}^{\infty} \mathcal{A}_{k n}(\delta)\, \mathbb{X}^{n} = \big( 1-\big(1+\delta\,(\xi+\xi^6)\big)^{k}\, \mathbb{X} \big)^{-1} + \big( 1-\big(1+\delta\,(\xi^2+\xi^5)\big)^{k}\, \mathbb{X} \big)^{-1} + {} \\ {} + \big( 1-\big(1+\delta\,(\xi^3+\xi^4)\big)^{k}\, \mathbb{X} \big)^{-1} = \frac{3 - 2\, \mathcal{A}_{k}(\delta) \, \mathbb{X} + \mathcal{B}_{k}(\delta)\, \mathbb{X}^2}{1- \mathcal{A}_{k}(\delta)\, \mathbb{X} + \mathcal{B}_{k}(\delta)\, \mathbb{X}^2 - (1-\delta-2\, \delta^2+\delta^3)^{k}\, \mathbb{X}^3}, \end{multline} and, in the sequel, for $k=1$ we get \begin{multline} \sum_{n=0}^{\infty} \mathcal{A}_{n}(\delta)\, \mathbb{X}^{n} = \big( 1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X} \big)^{-1} + \big( 1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X} \big)^{-1} + {} \\ {} + \big( 1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X} \big)^{-1} =\frac{\mathbb{X}^2\, p_{7}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)} = {} \\ {} = \frac{3+2\, (\delta-3)\, \mathbb{X} + (3-2\, \delta-2\, \delta^2)\, \mathbb{X}^2}{1+ (\delta-3)\, \mathbb{X} + (3-2\, \delta-2\, \delta^2)\, \mathbb{X}^2 + (-1+\delta+2\, \delta^2-\delta^3)\, \mathbb{X}^3}, \end{multline} where $p_{7}(\mathbb{X},\delta)=\mathbb{X}^3+(\delta -3)\mathbb{X}^2+ (3-2\delta -2\delta^2)\mathbb{X} + (-1+\delta +2\delta^2-\delta^3)$ (see~\cite{wsw1}). By~(\ref{w-bnd}) and~(\ref{w1.19}) we get (for~$k\in \mathbb{N}$): \begin{multline} \sum_{n=0}^{\infty} \mathcal{B}_{k n}(\delta)\, \mathbb{X}^{n} = \frac{\mathbb{X}^2\, r_{k}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, r_{k}\big( 1/\mathbb{X}; \delta\big)} = {} \\ {} = \frac{3-2\, \mathcal{B}_{k}(\delta)\, \mathbb{X} + (1-\delta-2\, \delta^2+\delta^3)\,\mathcal{A}_{k}(\delta)\, \mathbb{X}^2}{1- \mathcal{B}_{k}(\delta)\, \mathbb{X} + (1-\delta-2\, \delta^2+\delta^3)\,\mathcal{A}_{k}(\delta)\, \mathbb{X}^2 - (1-\delta-2\, \delta^2+\delta^3)^{2 k}\, \mathbb{X}^3}, \end{multline} where $r_{n}(\mathbb{X},\delta)$ is defined in~(\ref{w1.19}). By~(3.19) from~\cite{wsw1} we obtain \begin{multline} \sum_{n=0}^{\infty} {A}_{n}(\delta)\, \mathbb{X}^{n} = \frac{2-\xi^3-\xi^4}{1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X}} + \frac{2-\xi-\xi^6}{1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X}} + {}\\ {} + \frac{2-\xi^2-\xi^5}{1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X}} = \frac{1+(-2+\delta)\, \mathbb{X} + (1-\delta)\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)}; \end{multline} by~(3.18) and~(3.17) from~\cite{wsw1} we obtain respectively: \begin{multline} \sum_{n=0}^{\infty} {B}_{n}(\delta)\, \mathbb{X}^{n} = \frac{\xi+\xi^6-\xi^3-\xi^4}{1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X}} + \frac{\xi^2+\xi^5-\xi-\xi^6}{1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X}} + {} \\ {} + \frac{\xi^3+\xi^4-\xi^2-\xi^5}{1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X}} = \frac{\delta\, \mathbb{X} + (\delta^2-\delta)\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)} \end{multline} and \begin{equation} \sum_{n=0}^{\infty} {C}_{n}(\delta)\, \mathbb{X}^{n} = \sum_{n=0}^{\infty} \big(3 A_{n}(\delta) - B_{n}(\delta) - \mathcal{A}_{n}(\delta) \big)\, \mathbb{X}^{n} = \frac{\delta^2\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)}. \end{equation} % Equally, on the grounds of (\ref{w2.1})--(\ref{w2.3}) and (\ref{lem2.6-w6})--(\ref{lem2.6-w8}), we obtain the following formulas: \begin{equation} \sum_{n=0}^{\infty} a_{k n}\, \mathbb{X}^{n} = \frac{\sqrt{7}- (b_k + c_k )\,\mathbb{X}+ \widetilde{A}_{k} \,\mathbb{X}^2}{1- \mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3}, \end{equation} \begin{equation} \sum_{n=0}^{\infty} b_{k n}\, \mathbb{X}^{n} = \frac{\sqrt{7}- (a_k + c_k )\,\mathbb{X}+ \widetilde{B}_{k} \,\mathbb{X}^2}{1- \mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3}, \end{equation} \begin{equation} \sum_{n=0}^{\infty} c_{k n}\, \mathbb{X}^{n} = \frac{\sqrt{7}- (a_k + b_k )\,\mathbb{X}+ \widetilde{C}_{k} \,\mathbb{X}^2}{1- \mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3}. \end{equation} The special cases (for $k=1$) can also be generated in the following selective ways: \noindent by~(\ref{w2.1}) and~(\ref{w1-e}) we obtain \begin{multline} \sum_{n=0}^{\infty} a_{n}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{8\, \pi}{7}\big)\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{2\, \pi}{7}\big)\, \mathbb{X} } + \frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{4\, \pi}{7}\big)\, \mathbb{X} } = {}\\ {}= \frac{\sqrt{7}-2\, \sqrt{7}\,\mathbb{X}- \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} + 6\, \mathbb{X}^2-\mathbb{X}^3}; \end{multline} however, by~(\ref{w2.2}) and~(\ref{f2s-w1g}) we obtain \begin{multline} \sum_{n=0}^{\infty} b_{n}\, \mathbb{X}^{n} = \sum_{n=0}^{\infty} \big( a_{n+1} +2\, a_n \big) \, \mathbb{X}^{n} = \frac{1+2\, \mathbb{X}}{\mathbb{X}} \Big( \sum_{n=0}^{\infty} a_{n}\, \mathbb{X}^{n} \Big) -\frac{\sqrt{7}}{\mathbb{X}}={}\\ {}= \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{8\, \pi}{7}\big)\, \mathbb{X} } + \frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{2\, \pi}{7}\big)\, \mathbb{X} } + \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{4\, \pi}{7}\big)\, \mathbb{X} } = {}\\ {}= \frac{\sqrt{7}-3\, \sqrt{7}\,\mathbb{X}+3\, \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} + 6\, \mathbb{X}^2-\mathbb{X}^3} \end{multline} and \begin{equation} \sum_{n=0}^{\infty} c_{n}\, \mathbb{X}^{n} = \sum_{n=0}^{\infty} \big( a_n +2\, b_n - b_{n+1}\big) \, \mathbb{X}^{n} = \frac{\sqrt{7}-5\, \sqrt{7}\,\mathbb{X}+4\, \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} + 6\, \mathbb{X}^2-\mathbb{X}^3}. \end{equation} By~(\ref{wz6.8}) and~(\ref{w5n.9}) we get \begin{multline} \sum_{n=0}^{\infty} \Omega_{k n}(\delta)\, \mathbb{X}^{n} = \big( 1-\big(1+\delta\,(\xi-\xi^6)\big)^k\, \mathbb{X} \big)^{-1} + \big( 1-\big(1+\delta\,(\xi^2-\xi^5)\big)^k\, \mathbb{X} \big)^{-1} + {}\\ {}+\big( 1-\big(1+\delta\,(\xi^3-\xi^4)\big)^k\, \mathbb{X} \big)^{-1} = \frac{\mathbb{X}^2\, q_{k}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, q_{k}\big( 1/\mathbb{X}; \delta\big)} = {} \\ {} = \frac{3-2\, \Omega_{k}(\delta)\, \mathbb{X} + \tfrac{1}{2}\,\big( (\Omega_{k}(\delta))^{2} - \Omega_{2k}(\delta) \big)\, \mathbb{X}^2}{1- \Omega_{k}(\delta)\, \mathbb{X} + \tfrac{1}{2}\,\big( (\Omega_{k}(\delta))^{2} - \Omega_{2k}(\delta) \big)\, \mathbb{X}^2 - (1+i\, \sqrt{7}\,\delta+i\,\sqrt{7}\, \delta^3)^{k}\, \mathbb{X}^3}, \end{multline} and, the special case, for $k=1$ (by~(\ref{wz6.8}) and~(\ref{wz6.6})): \begin{multline} \sum_{n=0}^{\infty} \Omega_{n}(\delta)\, \mathbb{X}^{n} = \big( 1-\big(1+\delta\,(\xi-\xi^6)\big)\, \mathbb{X} \big)^{-1} + \big( 1-\big(1+\delta\,(\xi^2-\xi^5)\big)\, \mathbb{X} \big)^{-1} + {} \\ {} + \big( 1-\big(1+\delta\,(\xi^4-\xi^3)\big)\, \mathbb{X} \big)^{-1} =\frac{\mathbb{X}^2\, q_{1}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, q_{1}\big( 1/\mathbb{X}; \delta\big)} = {} \\ {} = \frac{3-2\, (3+i\, \sqrt{7}\,\delta)\, \mathbb{X} + (3+i\,2\,\sqrt{7}\,\delta)\, \mathbb{X}^2}{1- (3+i\, \sqrt{7}\,\delta)\, \mathbb{X} + (3+i\,2\,\sqrt{7}\,\delta)\, \mathbb{X}^2 - (1+i\, \sqrt{7}\,\delta+i\,\sqrt{7}\, \delta^3)\, \mathbb{X}^3}, \end{multline} where $q_{k}(\mathbb{X};\delta)$ is defined in~(\ref{w5n.9}). By~(\ref{w1-d}), (\ref{w2.11a})--(\ref{w2.11}) and~(\ref{lem2.6-w1})--(\ref{lem2.6-w3}) the following formulas may be obtained: \begin{multline}\label{w7-15} \sum_{n=0}^{\infty} f_{k n}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{-1-( g_{k} + h_{k} )\,\mathbb{X}+ {F}_{k}\,\mathbb{X}^2}{1- h_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2 -\mathbb{X}^3}, \end{multline} \begin{multline}\label{w7-16} \sum_{n=0}^{\infty} g_{k n}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{-1-( f_{k} + h_{k} )\,\mathbb{X}+ {G}_{k}\,\mathbb{X}^2}{1- h_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2 -\mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} h_{kn-1}\, \mathbb{X}^{n} = \frac{1}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{1}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{1}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{3-2\, h_{k-1}\,\mathbb{X}+ \tfrac{1}{2} \big( h_{k-1}^{2} - h_{2k-1} \big)\,\,\mathbb{X}^2}{1- h_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2 -\mathbb{X}^3} \stackrel{by~(\ref{wzor-gw1new})}{=} \frac{3-2\, {\cal B}_{k}\,\mathbb{X}+ (-1)^k\,{\cal A}_{k}\,\mathbb{X}^2}{1- {\cal B}_{k}\, \mathbb{X} + (-1)^k\,{\cal A}_{k}\, \mathbb{X}^2 -\mathbb{X}^3}. \end{multline} ($h_{-1}:=1$). We note that the special case of the formulas~(\ref{w7-15}) and~(\ref{w7-16}) for $k=1$ can be treated in another way (polynomial~(\ref{w1-d}) will be denoted here by~$p_7(\mathbb{X})$): \begin{multline} \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} = \frac{\xi+\xi^6}{1-(\xi+\xi^6)\, \mathbb{X}} + \frac{\xi^2+\xi^5}{1-(\xi^2+\xi^5)\, \mathbb{X}} + \frac{\xi^3+\xi^4}{1-(\xi^3+\xi^4)\, \mathbb{X}} = {}\\ {}= -\frac{\big( \mathbb{X}^3\,p_{7}\big( 1/\mathbb{X}\big) \big)^{'}}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}\big)} = \frac{-1+4\, \mathbb{X}+3\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3}, \end{multline} hence by~(\ref{f2s-w3g}): \begin{multline} \sum_{n=0}^{\infty} g_{n}\, \mathbb{X}^{n} = -1 + \sum_{n=1}^{\infty} \big( h_{n+1} - 2\, h_{n-1}\big)\, \mathbb{X}^{n} = {}\\ {} = -1 + \mathbb{X}^{-1}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} -\mathbb{X}^{-1}\, \big( h_{0}+h_{1}\, \mathbb{X}\big) -2\, \mathbb{X}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} = {}\\ {}= -6+\mathbb{X}^{-1} +\frac{1-2\, \mathbb{X}^2}{\mathbb{X}}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} = \frac{-1-3\, \mathbb{X}+3\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3} \end{multline} and, finally, using~(\ref{f2s-w3g}) again we get \begin{multline} \sum_{n=0}^{\infty} f_{n}\, \mathbb{X}^{n} = \sum_{n=0}^{\infty} \big( g_{n+1} - h_{n}\big)\, \mathbb{X}^{n} = {}\\ {} = \mathbb{X}^{-1}\, \sum_{n=0}^{\infty} g_{n}\, \mathbb{X}^{n} +\mathbb{X}^{-1} - \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} = -\frac{1+3\, \mathbb{X}+4\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3}. \end{multline} By~(\ref{w1-a}), (\ref{w4.4})--(\ref{w4.6}), (\ref{f2s-tw2-w-f}) and~(\ref{w4.19})--(\ref{w4.21}) we have \begin{multline} \sum_{n=0}^{\infty} x_{k n}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{\sqrt{7}-( y_{k} + z_{k} )\,\mathbb{X}+ x_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} y_{k n}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{\sqrt{7}-( x_{k} + z_{k} )\,\mathbb{X}+ y_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} z_{k n}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{\sqrt{7}-( x_{k} + y_{k} )\,\mathbb{X}+ z_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} and ($z_{-1}:=1$): \begin{multline} \sum_{n=0}^{\infty} z_{k n - 1}\, \mathbb{X}^{n} = \Big( 1- \big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} \Big)^{-1} + \Big( 1- \big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} \Big)^{-1} + {} \\ {} + \Big( 1- \big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} \Big)^{-1} = \frac{3-2\, z_{k-1}\,\mathbb{X}+ \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}. \end{multline} By~(\ref{w1-a}), (\ref{w1-d}), (\ref{w4.1})--(\ref{w4.3}), (\ref{f2s-tw2-w-f}), (\ref{w4.16a})--(\ref{w4.18}), (\ref{w4.30})--(\ref{w4.32}) and~(\ref{w4.6}), respectively, we get \begin{multline} \sum_{n=0}^{\infty} u_{k n}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{-1-( v_{k} + w_{k} )\,\mathbb{X}+ u_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} v_{k n}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{-1-( u_{k} + w_{k} )\,\mathbb{X}+ v_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} w_{k n}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } = {}\\ {} = \frac{-1-( u_{k} + v_{k} )\,\mathbb{X}+ w_{k}^{*}\,\mathbb{X}^2}{1- z_{k-1}\, \mathbb{X} + \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2 -\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} u_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{-1-( v_{k}^{*} + w_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, u_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} v_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{-1-( u_{k}^{*} + w_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, v_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} w_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{-1-( u_{k}^{*} + v_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, w_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}. \end{multline} By~(\ref{w1-a}), (\ref{w4.19})--(\ref{w4.21}), (\ref{w4.33})--(\ref{w4.35}) and~(\ref{w4.6}), we obtain \begin{multline} \sum_{n=0}^{\infty} x_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{\sqrt{7}-( y_{k}^{*} + z_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, x_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} y_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{\sqrt{7}-( x_{k}^{*} + z_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, y_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}, \end{multline} \begin{multline} \sum_{n=0}^{\infty} z_{k n}^{*}\, \mathbb{X}^{n} = \frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + \frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\ {}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } %= {}\\ {} = \frac{\sqrt{7}-( x_{k}^{*} + y_{k}^{*} )\,\mathbb{X}+ \big({-}\sqrt{7}\,\big)^{k}\, z_{k}\,\mathbb{X}^2}{1- \tfrac{1}{2} \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} + \big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2 -7^{k}\, \mathbb{X}^3}. \end{multline} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section 8 %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Jordan decomposition} For sequences $A_{n}(\delta)$, $B_{n}(\delta)$ and $C_{n}(\delta)$ we have the equality \begin{equation} \left[ \begin{array}{l} A_{n+1}(\delta) \\ B_{n+1}(\delta) \\ C_{n+1}(\delta) \\ \end{array} \right] = \mathcal{W}(\delta) \left[ \begin{array}{l} A_{n}(\delta) \\ B_{n}(\delta) \\ C_{n}(\delta) \\ \end{array} \right], \qquad n\in \mathbb{N}, \end{equation} where \begin{equation} \mathcal{W}(\delta)= \left[ \begin{array}{ccc} 1 & 2\, \delta & -\delta \\ \delta & 1 & 0 \\ 0 & \delta & 1-\delta \\ \end{array} \right]. \end{equation} Matrix $\mathcal{W}(\delta)$ is a~diagonalized matrix, and the following decomposition can be obtained \begin{equation} \mathcal{W}(\delta)= A\cdot \left[ \begin{array}{ccc} 1+\delta\, (\xi+\xi^6) & 0 & 0 \\ 0 & 1+\delta\, (\xi^2+\xi^5) & 0 \\ 0 & 0 & 1+\delta\, (\xi^3+\xi^4) \\ \end{array} \right] \cdot A^{-1}, \end{equation} where \begin{equation*} A = \left[ \begin{array}{ccc} 1+(\xi+\xi^6)^{-1} & 1+(\xi^2+\xi^5)^{-1} & 1+(\xi^3+\xi^4)^{-1} \\[1ex] (\xi+\xi^6)^{-2}+(\xi+\xi^6)^{-1} & (\xi^2+\xi^5)^{-2}+(\xi^2+\xi^5)^{-1} & (\xi^3+\xi^4)^{-2}+(\xi^3+\xi^4)^{-1} \\[1ex] (\xi+\xi^6)^{-2} & (\xi^2+\xi^5)^{-2} & (\xi^3+\xi^4)^{-2} \\ \end{array} \right] \end{equation*} and \begin{equation*} A^{-1} = \frac{1}{7} \left[ \begin{array}{ccc} (\xi^{\phantom{2}}+\xi^6)^{2} & (\xi^{\phantom{2}}+\xi^6)^{3} & (\xi^{\phantom{2}}+\xi^6)^{4} \\[1ex] (\xi^2+\xi^5)^{2} & (\xi^2+\xi^5)^{3} & (\xi^2+\xi^5)^{4}\\[1ex] (\xi^3+\xi^4)^{2} & (\xi^3+\xi^4)^{3} & (\xi^3+\xi^4)^{4}\\ \end{array} \right] \cdot \left[ \begin{array}{ccc} -3 & 2 & 5 \\[1ex] 1 & 1 & -2 \\[1ex] 2 & -1 & -2 \\ \end{array} \right]. \end{equation*} It should be noticed, that characteristic polynomial $w(\lambda)$ of matrix $\mathcal{W}(\delta)$ is equal to $$ w(\lambda) = -\delta^3\, p_{7} \Big( \frac{\lambda-1}{\delta} \Big). $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Tables} \renewcommand{\arraystretch}{1.3} \begin{table}[H] \begin{center} {\small \caption{}\label{f2s-tab1} \medskip \begin{tabular}{||c||r|r|r||r|r|r||r|r|r||}\hline\hline $n$ & \multicolumn{1}{|c|}{$a_{n}/\sqrt{7}$} & \multicolumn{1}{|c|}{$b_{n}/\sqrt{7}$} & \multicolumn{1}{|c||}{$c_{n}/\sqrt{7}$} & \multicolumn{1}{|c|}{$\alpha_{n}/\sqrt{7}$} & \multicolumn{1}{|c|}{$\beta_{n}/\sqrt{7}$} & \multicolumn{1}{|c||}{$\gamma_{n}/\sqrt{7}$} & \multicolumn{1}{|c|}{$f_{n}$} & \multicolumn{1}{|c|}{$g_{n}$} & \multicolumn{1}{|c||}{$h_{n}$} \\ \hline\hline $0 $ & $1 $ & $1 $ & $1 $ & $0 $ & $-2 $ & $1 $ & $-1 $ & $-1 $ & $-1 $ \\ \hline $1 $ & $3 $ & $2 $ & $0 $ & $-2 $ & $-5 $ & $3 $ & $-2 $ & $-2 $ & $5 $ \\ \hline $2 $ & $8 $ & $7 $ & $-2 $ & $-9 $ & $-15 $ & $8 $ & $-4 $ & $3 $ & $-4 $ \\ \hline $3 $ & $23 $ & $24 $ & $-9 $ & $-33 $ & $-47 $ & $23 $ & $-1 $ & $-8 $ & $13 $ \\ \hline $4 $ & $70 $ & $80 $ & $-33 $ & $-113 $ & $-150 $ & $70 $ & $-9 $ & $12 $ & $-16 $ \\ \hline $5 $ & $220 $ & $263 $ & $-113 $ & $-376 $ & $-483 $ & $220 $ & $3 $ & $-25 $ & $38 $ \\ \hline $6 $ & $703 $ & $859 $ & $-376 $ & $-1235 $ & $-1562 $ & $703 $ & $-22 $ & $41 $ & $-57 $ \\ \hline $7 $ & $2265 $ & $2797 $ & $-1235 $ & $-4032 $ & $-5062 $ & $2265 $ & $19 $ & $-79 $ & $117 $ \\ \hline $8 $ & $7327 $ & $9094 $ & $-4032 $ & $-13126 $ & $-16421 $ & $7327 $ & $-60 $ & $136 $ & $-193$ \\ \hline $9 $ & $23748 $ & $29547 $ & $-13126 $ & $-42673 $ & $-53295 $ & $23748 $ & $76 $ & $-253$ & $370 $ \\ \hline $10$ & $77043 $ & $95968 $ & $-42673 $ & $-138641$ & $-173011$ & $77043 $ & $-177$ & $446 $ & $-639$ \\ \hline $11$ & $250054$ & $311652$ & $-138641$ & $-450293$ & $-561706$ & $250054$ & $269 $ & $-816$ & $1186$ \\ \hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table}[H] \begin{center} {\small \caption{}\label{f2s-tab7} \medskip \begin{tabular}{||c||r|r|r|r|r|r|r|r|r|r|r|r||}\hline\hline $n$ & $0 $ & $1 $ & $2 $ & $3 $ & $4 $ & $5 $ & $6 $ & $7 $ & $8 $ & $9 $ & $10$ & $11$ \\ \hline\hline $\varepsilon_{n}$ & $1 $ & $0 $ & $-2 $ & $2 $ & $2 $ & $-6 $ & $2 $ & $10 $ & $-14$ & $-6 $ & $34 $ & $-22$ \\ \hline $\omega_{n}$ & $0 $ & $1 $ & $-1 $ & $-1 $ & $3 $ & $-1 $ & $-5 $ & $7 $ & $3 $ & $-17$ & $11 $ & $23 $ \\ \hline $\Psi_{n}$ & $2 $ & $-1 $ & $-3 $ & $5 $ & $1 $ & $-11 $ & $9 $ & $13 $ & $-31 $ & $5 $ & $57 $ & $-67 $ \\ \hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table}[H] \begin{center} {\small \caption{}\label{f2s-tab3} \medskip \begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline $n$ & \multicolumn{1}{|c|}{$u_{n}$} & \multicolumn{1}{|c|}{$v_{n}$} & \multicolumn{1}{|c||}{$w_{n}$} & \multicolumn{1}{|c|}{$x_{n}$} & \multicolumn{1}{|c|}{$y_{n}$} & \multicolumn{1}{|c||}{$z_{n}$} \\ \hline\hline $0 $ & $-1 $ & $-1 $ & $-1 $ & $\sqrt{7} $ & $\sqrt{7} $ & $\sqrt{7} $ \\ \hline $1 $ & $\sqrt{7} $ & $-2 \sqrt{7} $ & $0 $ & $0 $ & $0 $ & $7 $ \\ \hline $2 $ & $0 $ & $-7 $ & $0 $ & $\sqrt{7} $ & $2 \sqrt{7} $ & $4 \sqrt{7} $ \\ \hline $3 $ & $\sqrt{7} $ & $-6 \sqrt{7} $ & $\sqrt{7} $ & $0 $ & $7 $ & $21 $ \\ \hline $4 $ & $0 $ & $-28 $ & $7 $ & $0 $ & $7 \sqrt{7} $ & $14 \sqrt{7} $ \\ \hline $5 $ & $0 $ & $-21 \sqrt{7} $ & $7 \sqrt{7} $ & $-7 $ & $35 $ & $70 $ \\ \hline $6 $ & $-7 $ & $-105 $ & $42 $ & $-7 \sqrt{7} $ & $28 \sqrt{7} $ & $49 \sqrt{7} $ \\ \hline $7 $ & $-7 \sqrt{7} $ & $-77 \sqrt{7} $ & $35 \sqrt{7} $ & $-49 $ & $147 $ & $245 $ \\ \hline $8 $ & $-49 $ & $-392 $ & $196 $ & $-42 \sqrt{7} $ & $112 \sqrt{7}$ & $175 \sqrt{7}$ \\ \hline $9 $ & $-42 \sqrt{7} $ & $-287 \sqrt{7} $ & $154 \sqrt{7}$ & $-245 $ & $588 $ & $882 $ \\ \hline $10$ & $-245 $ & $-1470 $ & $833 $ & $-196 \sqrt{7}$ & $441 \sqrt{7}$ & $637 \sqrt{7}$ \\ \hline $11$ & $-196 \sqrt{7}$ & $-1078 \sqrt{7}$ & $637 \sqrt{7}$ & $-1078 $ & $2303 $ & $3234 $ \\ \hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table}[H] \begin{center} {\small \caption{}\label{f2s-tab2} \medskip \begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline $n$ & \multicolumn{1}{|c|}{$u_{n}^{*}$} & \multicolumn{1}{|c|}{$v_{n}^{*}$} & \multicolumn{1}{|c||}{$w_{n}^{*}$} & \multicolumn{1}{|c|}{$x_{n}^{*}/\sqrt{7}$} & \multicolumn{1}{|c|}{$y_{n}^{*}/\sqrt{7}$} & \multicolumn{1}{|c||}{$z_{n}^{*}/\sqrt{7}$} \\ \hline\hline $0 $ & $-1 $ & $-1 $ & $-1 $ & $1 $ & $1 $ & $1 $ \\ \hline $1 $ & $-7 $ & $7 $ & $0 $ & $1 $ & $2 $ & $-3 $ \\ \hline $2 $ & $-14 $ & $7 $ & $-7 $ & $7 $ & $7 $ & $0 $ \\ \hline $3 $ & $-56 $ & $42 $ & $-7 $ & $14 $ & $21 $ & $-14 $ \\ \hline $4 $ & $-147 $ & $98 $ & $-49 $ & $56 $ & $63 $ & $-21 $ \\ \hline $5 $ & $-490 $ & $343 $ & $-98 $ & $147 $ & $196 $ & $-98 $ \\ \hline $6 $ & $-1421 $ & $980 $ & $-392 $ & $490 $ & $588 $ & $-245 $ \\ \hline $7 $ & $-4459 $ & $3087 $ & $-1029 $ & $1421 $ & $1813 $ & $-833 $ \\ \hline $8 $ & $-13377 $ & $9261 $ & $-3430 $ & $4459 $ & $5488 $ & $-2401 $ \\ \hline $9 $ & $-41160 $ & $28469 $ & $-9947 $ & $13377 $ & $16807 $ & $-7546 $ \\ \hline $10$ & $-124852$ & $86436 $ & $-31213$ & $41160 $ & $51107 $ & $-22638 $ \\ \hline $11$ & $-381759$ & $264110$ & $-93639$ & $124852 $ & $156065 $ & $-69629 $ \\ \hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table}[H] \begin{center} {\small \caption{}\label{f2s-tab6} \medskip \begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline $n$ & \multicolumn{1}{|c|}{$F_{n}$} & \multicolumn{1}{|c|}{$G_{n}$} & \multicolumn{1}{|c||}{$H_{n}$} & \multicolumn{1}{|c|}{$\widetilde{A}_{n}/\sqrt{7}$} & \multicolumn{1}{|c|}{$\widetilde{B}_{n}/\sqrt{7}$} & \multicolumn{1}{|c||}{$\widetilde{C}_{n}/\sqrt{7}$} \\ \hline\hline $0 $ & $-1 $ & $-1 $ & $-1 $ & $1 $ & $1 $ & $1 $ \\ \hline $1 $ & $-4 $ & $3 $ & $3 $ & $-1 $ & $3 $ & $4 $ \\ \hline $2 $ & $5 $ & $-9 $ & $-2 $ & $-8 $ & $15 $ & $19 $ \\ \hline $3 $ & $-15 $ & $20 $ & $6 $ & $-42 $ & $76 $ & $95 $ \\ \hline $4 $ & $31 $ & $-46 $ & $-11 $ & $-213 $ & $384 $ & $479 $ \\ \hline $5 $ & $-72 $ & $103 $ & $26 $ & $-1076 $ & $1939 $ & $2418 $ \\ \hline $6 $ & $160 $ & $-232 $ & $-57 $ & $-5433 $ & $9790 $ & $12208 $ \\ \hline $7 $ & $-361 $ & $521 $ & $129 $ & $-27431 $ & $49429 $ & $61637 $ \\ \hline $8 $ & $810 $ & $-1171$ & $-289 $ & $-138497 $ & $249563 $ & $311200 $ \\ \hline $9 $ & $-1821$ & $2631 $ & $650 $ & $-699260 $ & $1260023 $ & $1571223 $ \\ \hline $10$ & $4091 $ & $-5912$ & $-1460$ & $-3530506 $ & $6361752 $ & $7932975 $ \\ \hline $11$ & $-9193$ & $13284$ & $3281 $ & $-17825233 $ & $32119960 $ & $40052935 $ \\ \hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\arraystretch}{1.3} \begin{table}[H] {\small %scriptsize \caption{}\label{f2s-tab5} \medskip \mbox{\null}\hfill \begin{tabular}{||c||l||} \hline\hline $n$ & \multicolumn{1}{|c||}{$\Omega_n(\delta )$} \\ \hline\hline $0$ & $3 $ \\ \hline $1$ & $i \sqrt{7} \delta +3 $ \\ \hline $2$ & $-7 \delta^2+2 i \sqrt{7} \delta +3 $ \\ \hline $3$ & $-4 i \sqrt{7} \delta^3-21 \delta^2+3 i \sqrt{7}\delta +3 $ \\ \hline $4$ & $21 \delta^4-16 i \sqrt{7} \delta^3-42 \delta^2+4 i \sqrt{7} \delta +3 $ \\ \hline $5$ & $14 i \sqrt{7} \delta^5+105 \delta^4-40 i \sqrt{7} \delta^3-70 \delta^2+5 i \sqrt{7} \delta +3 $ \\ \hline $6$ & $-70 \delta^6+84 i \sqrt{7} \delta^5+315 \delta^4-80 i \sqrt{7} \delta^3-105\delta^2+6 i \sqrt{7} \delta +3 $ \\ \hline $7$ & $-49 i \sqrt{7} \delta^7-490 \delta^6+294 i \sqrt{7} \delta^5+735 \delta^4-140 i\sqrt{7} \delta^3-147 \delta^2+7 i\sqrt{7} \delta +3 $ \\ \hline\hline \end{tabular} \hfill\mbox{\null} } \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\arraystretch}{1.3} \begin{table}[H] {\scriptsize \caption{}\label{f2s-tab4} \medskip \mbox{\null}\hfill \begin{tabular}{||c|l|l||} \hline\hline $n$ & \multicolumn{2}{|c||}{$p_n(\delta )$} \\ \hline\hline $0$ & \multicolumn{2}{|l||}{$1 $} \\ \hline $1$ & \multicolumn{2}{|l||}{$1 $} \\ \hline $2$ & \multicolumn{2}{|l||}{$-2 \delta^2 + 1 $} \\ \hline $3$ & \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^3-6\delta^2+1 $} \\ \hline $4$ & \multicolumn{2}{|l||}{$3 \delta^4-4 i \sqrt{7} \delta^3-12 \delta ^2+1 $} \\ \hline $5$ & \multicolumn{2}{|l||}{$5 i \sqrt{7} \delta^5+15 \delta^4-10 i \sqrt{7} \delta^3-20 \delta^2+1 $} \\ \hline $6$ & \multicolumn{2}{|l||}{$-8 \delta^6+30 i \sqrt{7} \delta^5+45 \delta^4-20 i \sqrt{7} \delta^3-30 \delta^2+1 $} \\ \hline $7$ & \multicolumn{2}{|l||}{$-17 i \sqrt{7} \delta^7-56 \delta^6+105 i \sqrt{7} \delta^5+105 \delta^4-35 i \sqrt{7}\delta^3-42 \delta^2+1$} \\ \hline\hline $n$ & \multicolumn{2}{|c||}{$r_n(\delta )$} \\ \hline\hline $0$ & \multicolumn{2}{|l||}{$0 $} \\ \hline $1$ & \multicolumn{2}{|l||}{$\delta $} \\ \hline $2$ & \multicolumn{2}{|l||}{$2 \delta $} \\ \hline $3$ & \multicolumn{2}{|l||}{$-2 \delta^3+3 \delta $} \\ \hline $4$ & \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^4-8 \delta^3+4 \delta $} \\ \hline $5$ & \multicolumn{2}{|l||}{$3 \delta^5-5 i \sqrt{7} \delta^4-20 \delta^3+5 \delta $} \\ \hline $6$ & \multicolumn{2}{|l||}{$5 i \sqrt{7} \delta^6+18 \delta^5-15 i \sqrt{7} \delta^4-40 \delta^3+6 \delta $} \\ \hline $7$ & \multicolumn{2}{|l||}{$-8 \delta^7+35 i \sqrt{7} \delta^6+63 \delta^5-35 i \sqrt{7} \delta^4-70 \delta^3+7 \delta$} \\ \hline\hline $n$ & \multicolumn{1}{|c|}{$s_n(\delta )$} & \multicolumn{1}{|c||}{$k_n(\delta )$} \\ \hline\hline $0$ & $0 $ & $0$ \\ \hline $1$ & $0 $ & $0 $ \\ \hline $2$ & $0 $ & $0 $ \\ \hline $3$ & $\delta^3 $ & $0 $ \\ \hline $4$ & $4 \delta^3 $ & $-2 \delta^4 $ \\ \hline $5$ & $-5 \delta^5+10 \delta^3 $ & $-10 \delta^4 $ \\ \hline $6$ & $-i \sqrt{7} \delta^6-30 \delta^5+20 \delta^3 $ & $10 \delta^6-30 \delta^4 $ \\ \hline $7$ & $18 \delta^7-7 i \sqrt{7} \delta^6-105 \delta^5+35 \delta^3$ & $2 i \sqrt{7} \delta^7+70 \delta^6-70 \delta^4$ \\ \hline\hline $n$ & \multicolumn{2}{|c||}{$l_n(\delta )$} \\ \hline\hline $0$ & \multicolumn{2}{|l||}{$0 $} \\ \hline $1$ & \multicolumn{2}{|l||}{$0 $} \\ \hline $2$ & \multicolumn{2}{|l||}{$\delta^2 $} \\ \hline $3$ & \multicolumn{2}{|l||}{$3 \delta^2 $} \\ \hline $4$ & \multicolumn{2}{|l||}{$-3 \delta^4 + 6 \delta^2 $} \\ \hline $5$ & \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^5-15 \delta^4+10 \delta^2 $} \\ \hline $6$ & \multicolumn{2}{|l||}{$8 \delta^6-6 i \sqrt{7} \delta^5-45 \delta^4+15 \delta^2 $} \\ \hline $7$ & \multicolumn{2}{|l||}{$6 i \sqrt{7} \delta^7+56 \delta^6-21 i \sqrt{7} \delta^5-105 \delta^4+21 \delta^2$} \\ \hline\hline \end{tabular} \hfill\mbox{\null} } \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Acknowledgments} The authors wish to express their gratitude to the referee for several helpful comments and suggestions concerning the first version of our paper, especially for preparing Example~\ref{example2.2}, which is added to the present version of the paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{10} \bibitem{bradley} D.~M.~Bradley, {A~class of series acceleration formulae for Catalan's constant}, \textit{Ramanujan~J.} \textbf{3} (1999), 149--173. \bibitem{GrzymkowskiWitula} R.~Grzymkowski and R.~Witu{\l}a, \textit{Calculus Methods in Algebra, Part One}, WPKJS, 2000 (in Polish). %Gliwice \bibitem{Ireland} K.~Ireland, M.~Rosen, \textit{A~Classical Introduction to Modern Number Theory}, Springer Verlag, 1982. %New York \bibitem{Shevelev} V.~S.~Shevelev, {Three Ramanujan's formulas}, \textit{Kvant} \textbf{6} (1988), 52--55 (in Russian). \bibitem{ShklyarskyUSA} %Problem 230 D.~O.~Shklyarsky, N.~N.~Chentzov, I.~M.~Yaglom, \textit{The USSR Olympiad Problem Book}, W.~H.~Freeman \& Co., 1962. %San Francisco \bibitem{ShklyarskyMir} %Problem 329 D.~O.~Shklyarsky, N.~N.~Chentsov, I.~M.~Yaglom, \textit{Selected Problems and Theorems in Elementary Mathematics, Arithmetic and Algebra}, Mir, 1979. %Moscow \bibitem{sloan} N.~J.~A. Sloane, \htmladdnormallink{\textit{The On-Line Encyclopedia of Integer Sequences}} {http://www.research.att.com/$\sim$njas/sequences/}, 2007. \bibitem{surowski} D.~Surowski and P.~McCombs, {Homogeneous polynomials and the minimal polynomial of $\cos(2\pi/n)$}, \textit{Missouri J.~Math. Sci.} \textbf{15} (2003), 4--14. \bibitem{Watkins} W.~Watkins and J.~Zeitlin, {The minimal polynomials of $\cos (2\pi /n)$}, \textit{Amer. Math. Monthly} \textbf{100} (1993), 471--474. \bibitem{wsw1} R.~Witu{\l}a, D.~S{\l}ota and A.~Warzy{\'n}ski, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html}{Quasi-Fibonacci numbers of the seventh order}, \textit{J.~Integer Seq.} \textbf{9} (2006), Article 06.4.3. \bibitem{Yaglom} A.~M.~Yaglom and I.~M.~Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for $\pi$, \textit{Uspekhi Matem. Nauk} \textbf{VIII} (1953), 181{--}187 (in Russian). %no. 5 \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11B83, 11A07; Secondary 39A10. \noindent \emph{Keywords: } Ramanujan's formulas, Fibonacci numbers, primitive roots of unity, recurrence relation. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A001607}, \seqnum{A002249}, \seqnum{A079309}, \seqnum{A094648}, \seqnum{A110512}, and \seqnum{A115146}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 19 2007; revised version received May 17 2007. Published in {\it Journal of Integer Sequences}, June 4 2007. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in \end{document}