\documentclass[12pt,reqno]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{epsf} \usepackage{fullpage} \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} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \vspace*{1cm} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \mbox{\null} \hspace*{4cm} \end{center} \begin{center} \vskip 1cm{\LARGE\bf Ramanujan Type Trigonometric Formulas:\\ \vskip .1in The General Form for the Argument $\displaystyle\frac{2\, \pi}{7}$} \vskip 1cm \large Roman Witu{\l}a \\ Institute of Mathematics \\ Silesian University of Technology \\ Kaszubska 23 \\ Gliwice 44-100 \\ Poland \\ \href{mailto:roman.witula@polsl.pl}{\tt roman.witula@polsl.pl} \\ \end{center} \vskip .2in \begin{abstract} In this paper, we present many general identities connected with the classical Ramanujan equality. Moreover, we give Binet formulas for an accelerator sequence for Catalan's constant. \end{abstract} \vskip .2in \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{defin}[theorem]{Definition} \newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}} \newtheorem{examp}[theorem]{Example} \newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}} \newtheorem{rema}[theorem]{Remark} \newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}} \section{Introduction} The main objective of this paper is to obtain some general trigonometric formulas related to the known Ramanujan equality~\cite{Berndt,Berndt2,Berndt3,Kreczmar,Shevelev1988}: \begin{equation}\label{w-1.0} \sqrt[3]{\cos \alpha} + \sqrt[3]{\cos 2\alpha} + \sqrt[3]{\cos 4\alpha} = \sqrt[3]{\tfrac{1}{2} \big( 5 - 3\, \sqrt[3]{7} \big)}, \end{equation} where $\alpha=2 \pi/7$. Other formulas of this type, referring to the ninth and eleventh primitive roots of unity, etc., will be published in separate papers. The present paper, in a way, is a~continuation of previous papers~\cite{WitulaSlota-A7,WitulaSlota-A7s,WitulaSlota-A11} and I will take advantage of some results from those papers. The quasi-Fibonacci sequences of the seventh order discussed in the above mentioned papers are applied here for describing some attractive formulas involving radicals. The paper is divided into five parts: \begin{list}{}{} \item[--] Section 2~-- where some striking equalities related to equality~(\ref{w-1.0}) are presented. Furthermore, Binet formulas for two new sequences $\{\mathcal{S}_{n}\}$ and $\{\mathcal{S}_{n}^{*}\}$ are derived, which, at the same time, resolves the problem of an algebraic description of the zeros of polynomials $x^3-\sqrt[3]{7}x-1$ and $x^3-\sqrt[3]{49}x-1$ (see Remark~\ref{rem2-1}). \item[--] Section 3~-- where the fundamental formula~(\ref{ram-gw}) for a~sum of the cube roots of the three roots of a cubic polynomial is given. \item[--] Section 4~-- where many basic sequences of integers, reals and complex numbers, introduced and discussed earlier by the authors in papers~\cite{WitulaSlota-A7,WitulaSlota-A7s,WitulaSlota-A11}, are presented. In addition, some new relations between the elements of the sequences are discussed. Furthermore, a~new description of Binet's formula is introduced for an accelerator sequence for Catalan's constant, which, naturally, makes it possible to extend this formula for all integers (see Remark~\ref{rem4-4}). \item[--] Section 5~-- where applications of the formula~(\ref{ram-gw}) to many special kinds of polynomial of degree three are given. This section contains many Ramanujan type trigonometric formulas. Moreover in Remark~\ref{rem4.3} the nontrivial theoretical discussion of some numerical case is presented. \end{list} \section{Delicious} Now we are going to prove the following three interesting identities: \begin{multline}\label{w-nr2} \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\, \big( 2\, \cos \alpha \big)^{k} + \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \big( 2\, \cos 2\alpha \big)^{k} %+ {}\\ {}+ \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\, \big( 2\, \cos 4\alpha \big)^{k} ={}\\ {}= \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\, \big( 2\, \cos 2\alpha \big)^{k+1} + \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \big( 2\, \cos 4\alpha \big)^{k+1} + {}\\ {}+ \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\, \big( 2\, \cos \alpha \big)^{k+1} %={}\\ {}= \sqrt[3]{7}\, \psi_{k}, \end{multline} where $ \psi_{0}=-1$, $ \psi_{1}=0$, $ \psi_{2}=-3$ and $$ \psi_{k+3} + \psi_{k+2} -2\, \psi_{k+1} - \psi_{k} =0, \qquad k\in \mathbb{Z}; $$ % \begin{multline}\label{w-nr1} \sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\, \big( 2\, \cos \alpha \big)^{k} + \sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\, \big( 2\, \cos 2\alpha \big)^{k} + \sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\, \big( 2\, \cos 4\alpha \big)^{k} ={}\\ {}= \sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\, \big( 2\, \cos \alpha \big)^{k+1} + \sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\, \big( 2\, \cos 2\alpha \big)^{k+1} + {}\\ {}+ \sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\, \big( 2\, \cos 4\alpha \big)^{k+1} {}= \sqrt[3]{49}\, \varphi_{k}, \end{multline} where $ \varphi_{0}=0$, $ \varphi_{1}=-1$, $ \varphi_{2}=1$ and $$ \varphi_{k+3} + \varphi_{k+2} -2\, \varphi_{k+1} - \varphi_{k} =0, \qquad k\in \mathbb{Z}; $$ \begin{multline}\label{w-nr3} \sqrt[3]{\sec 2\alpha}\, \big( 2\, \cos \alpha \big)^{k} + \sqrt[3]{\sec 4\alpha}\, \big( 2\, \cos 2\alpha \big)^{k} + \sqrt[3]{\sec \alpha}\, \big( 2\, \cos 4\alpha \big)^{k} ={}\\ {}= \delta_k\, \sqrt[3]{8-6\, \sqrt[3]{7}} + \sigma_k\, \sqrt[3]{6\, \big( 1+\sqrt[3]{7}\big)^2} + \xi_k\, \sqrt[3]{2\, \big( 5-3\,\sqrt[3]{7}\big)^2} ={}\\ {}= \sqrt[3]{ f_{3k+1} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{3k+1,8} + \sqrt{\mathcal{T}_{3k+1,8}} } + \sqrt[3]{ \mathcal{S}_{3k+1,8} - \sqrt{\mathcal{T}_{3k+1,8}} }\,\, \Big)}, \end{multline} where \begin{align*} \delta_0 &= 1, & \delta_1 &= 0, & \delta_2 &= 0,\\ \sigma_0 &= 0, & \sigma_1 &= -1, & \sigma_2 &= 0,\\ \xi_0 &= 0, & \xi_1 &= 0, & \xi_2 &= 1, \end{align*} $$ \mathbb{X}_{k+3} + \mathbb{X}_{k+2} -2\, \mathbb{X}_{k+1} - \mathbb{X}_{k} =0, \qquad k=0,1,2,\ldots, $$ for every $\mathbb{X}\in \{\delta,\sigma,\xi\}$, whereas the sequences $f_{3k+1}$, $\mathcal{S}_{3k+1,8}$ and $\mathcal{T}_{3k+1,8}$ are defined by formulas~(\ref{f2s-w3g}), (\ref{w6-e}) and~(\ref{w6-f}) (in Section~\ref{roz3}, other sequences occurring in the definition of sequences $\mathcal{S}_{3k+1,8}$ and $\mathcal{T}_{3k+1,8}$ are defined as well). The first twelve values of numbers $\psi_n$ and $\varphi_n$ are presented in Table~\ref{tabela}. Moreover an interesting numerical link to the formula~(\ref{w-nr2}) are the considerations from Remark~\ref{rem4.3}. \medskip\noindent \textit{Proof of formulas~(\ref{w-nr2})--(\ref{w-nr3}).} For $k=0,1,2$, the formulas (\ref{w-nr2})--(\ref{w-nr3}) follow from~(\ref{sf-w2}), (\ref{sf-w5}), (\ref{sf-w7a}), (\ref{sf-w8}) and (\ref{sf-w11}), and (or) from the following equalities (in both cases, equality~(\ref{w-0}) below for $\mathbb{X}=0$ is needed): \begin{multline*} \sqrt[3]{\frac{\cos\alpha}{\cos 2\alpha}}\, \big( 2\, \cos\alpha \big)^2 + \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \big( 2\, \cos 2\alpha \big)^2 + \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\, \big( 2\, \cos 4\alpha \big)^2 = {}\\ {}= \sqrt[3]{\frac{\cos\alpha}{\cos 2\alpha}}\, \big( 2+2\, \cos 2\alpha \big) {}+ \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \big( 2+2\, \cos 4\alpha \big) + \sqrt[3]{\frac{\cos 4\alpha}{\cos\alpha}}\, \big( 2+2\, \cos\alpha \big) ={}\\ %\end{multline*} %\begin{multline*} {}= 2\, \sqrt[3]{7} \, \psi_0 + 2\, \Big( \sqrt[3]{\cos \alpha\, \cos^2 2\alpha} + \sqrt[3]{\cos 2\alpha\, \cos^2 4\alpha} + \sqrt[3]{\cos 4\alpha\, \cos^2 \alpha}\, \Big) = {}\\ {}= -2\, \sqrt[3]{7} + \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}} + \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}} + \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}} = -2\, \sqrt[3]{7} + \sqrt[3]{7} \, \psi_0 = -3\, \sqrt[3]{7}, \end{multline*} and \begin{multline*}%equation*} \sqrt[3]{\frac{\cos\alpha}{\cos 4\alpha}}\, \big( 2\, \cos \alpha \big)^2 + \sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\, \big( 2\, \cos 2\alpha \big)^2 + \sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\, \big( 2\, \cos 4\alpha \big)^2 ={}\\ {}= \sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}\, \big( 2 + 2\, \cos 2\alpha \big) + \sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}}\, \big( 2 + 2\, \cos 4\alpha \big) + \sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}}\, \big( 2 + 2\, \cos \alpha \big) ={}%\\ \end{multline*}%equation*} %\pagebreak \begin{multline*} {}=2\, \sqrt[3]{49}\, \varphi_0 + \left(\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\right)^2+ \left(\sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\right)^2+ \left(\sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\right)^2 ={}\\ =\!\! \left(\sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}} + \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}+ \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\right)^{\!\!2} -2\! \left( \sqrt[3]{\frac{\cos 2\alpha}{\cos \alpha}} + \sqrt[3]{\frac{\cos \alpha}{\cos 4\alpha}}+ \sqrt[3]{\frac{\cos 4\alpha}{\cos 2\alpha}} \right) \!\!=\\ {}= \Big( \sqrt[3]{7}\, \psi_0 \Big)^2 -2\,\sqrt[3]{49}\, \varphi_0=\sqrt[3]{49}. \end{multline*} Now let us set $$ \mathfrak{B}_n := \sum_{k=0}^{2} x_k\, \big( \cos \big(2^{k}\alpha\big) \big)^n, $$ where $x_k\in \mathbb{R}$, $k=1,2,3$, are given. Then, from Newton's formula we obtain $$ \mathfrak{B}_{n+3}+\mathfrak{B}_{n+2}-2\,\mathfrak{B}_{n+1} -\mathfrak{B}_{n} =0 $$ since~ \cite{Kreczmar,WitulaSlota-A7,WitulaSlota-A7s}: \begin{equation}\label{w-0} \prod_{k=0}^{2} \Big( \mathbb{X} - 2\, \cos \big(2^{k}\alpha \big) \Big) = \mathbb{X}^3 + \mathbb{X}^2 -2\, \mathbb{X} -1. \end{equation} Hence, on account of the definitions of sequences $\varphi_k$ and $\psi_k$, $k\in \mathbb{N}$, by applied induction arguments the formulas~(\ref{w-nr2}) and~(\ref{w-nr1}) follow. Similarly, by applying ~(\ref{sf-w1}), (\ref{sf-w4}) and~(\ref{sf-w7}) we deduce the first part of~(\ref{w-nr3}). The second part of~(\ref{w-nr3}) from~(\ref{w6-d}) follows. \begin{remark}\label{rem2-1} Let $a,b,c\in \mathbb{C}$ and $a+b+c=0$. Put $$ s_k := a^k + b^k + c^k, \qquad k\in \mathbb{N}. $$ Then the following relations hold \cite{Kreczmar,Modenov}: \begin{align*} 2\, s_4 &= s_{2}^{2}; & 6\, s_5 &=5\, s_2\, s_3; \\ 6\, s_7 &= 7\, s_3\, s_4; & 10\, s_7 &= 7\, s_2\, s_5;\\ 25\, s_3\, s_7 &= 21 s_{5}^{2}; & 50\, s_{7}^{2} &= 49\, s_4\, s_{5}^{2} \end{align*} and the respective Newton formula has the form $$ s_{n+3} = a\, b\, c\, s_n + \frac{1}{2}\, s_2\, s_{n+1}, \qquad n\in \mathbb{N}. $$ Hence, and from~(\ref{w-nr1}) for $k=0$, we get $$ \mathcal{S}_{n+3} = \sqrt[3]{7}\, \mathcal{S}_{n+1} + \mathcal{S}_n , $$ where $\mathcal{S}_0=3$, $\mathcal{S}_1=0$, $\mathcal{S}_2=2\,\sqrt[3]{7}$, $$ \mathcal{S}_n := \left(\frac{\cos \alpha}{\cos 4\alpha}\right)^{\!n/3}+ \left(\frac{\cos 2\alpha}{\cos \alpha}\right)^{\!n/3}+ \left(\frac{\cos 4\alpha}{\cos 2\alpha}\right)^{\!n/3} $$ which implies the following formula $$ \mathcal{S}_n = \widehat{a}_n+ \widehat{b}_n\, \sqrt[3]{7} + \widehat{c}_n\, \sqrt[3]{49}, $$ where \begin{align*} \widehat{a}_{n+3} &= \widehat{a}_n +7\, \widehat{c}_n,\\ \widehat{b}_{n+3} &= \widehat{b}_n + \widehat{a}_{n+1},\\ \widehat{c}_{n+3} &= \widehat{c}_n + \widehat{b}_{n+1},\\ \widehat{a}_0 &=3, \quad \widehat{a}_1=0,\quad \widehat{a}_2=0,\\ \widehat{b}_0 &=0, \quad \widehat{b}_1=0,\quad \widehat{b}_2=1,\\ \widehat{c}_0 &=0, \quad \widehat{c}_1=0,\quad \widehat{c}_2=0. \end{align*} On the other hand, from~(\ref{w-nr2}) for $k=1$, we obtain $$ \mathcal{S}_{n+3}^{*} = \frac{1}{2}\, \mathcal{S}_{2}^{*}\, \mathcal{S}_{n+1}^{*} + \mathcal{S}_{n}^{*} , \qquad n\in \mathbb{N}_0, $$ where $\mathcal{S}_{0}^{*}=3$, $\mathcal{S}_{1}^{*}=0$, \begin{equation*} %\begin{multline*} \mathcal{S}_{n}^{*} := \bigg(2\,\cos \alpha\, \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}\, \bigg)^{\!\!n}+ \bigg(2\,\cos 2\alpha\, \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}\, \bigg)^{\!\!n} %+{}\\ %{} + \bigg(2\,\cos 4\alpha\, \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}\, \bigg)^{\!\!n}, \quad n\in \mathbb{N}, %\end{multline*} \end{equation*} and by~(\ref{w-0}) for $\mathbb{X}=0$: \begin{multline*} \mathcal{S}_{2}^{*} := \left( \frac{\cos \alpha}{\cos 2\alpha} \right)^{\!\!2/3} \big( 2+2\,\cos 2\alpha\big) + \left( \frac{\cos 2\alpha}{\cos 4\alpha} \right)^{\!\!2/3}\!\! \big( 2+2\,\cos 4\alpha\big) %+{}\\ {}+ \left( \frac{\cos 4\alpha}{\cos \alpha} \right)^{\!\!2/3}\!\! \big( 2+2\,\cos \alpha \big) ={}\\ {}= 2\, \left( \left(\frac{\cos \alpha}{\cos 2\alpha} \right)^{\!\!1/3} + \left(\frac{\cos 2\alpha}{\cos 4\alpha} \right)^{\!\!1/3} + \left(\frac{\cos 4\alpha}{\cos \alpha} \right)^{\!\!1/3}\, \right)^{\!\!2} -{}\\ {} -4\, \left( \left(\frac{\cos \alpha}{\cos 4\alpha} \right)^{\!\!1/3} + \left(\frac{\cos 4\alpha}{\cos 2\alpha} \right)^{\!\!1/3} + \left(\frac{\cos 2\alpha}{\cos \alpha} \right)^{\!\!1/3}\, \right) +{}\\ {}+ \big( 2\, \cos \alpha \big)^{1/3}\, \big( 4\, \cos \alpha\, \cos 2\alpha \big)^{1/3} + \big( 2\, \cos 2\alpha \Big)^{1/3}\, \big( 4\, \cos 2\alpha\, \cos 4\alpha \big)^{1/3} +{}\\ {}+ \big( 2\, \cos 4\alpha \big)^{1/3}\, \big( 4\, \cos \alpha\, \cos 4\alpha \big)^{1/3} = 2\, (\sqrt[3]{7}\,\psi_0)^2 - 4\, \mathcal{S}_1 + \mathcal{S}_1 = 2\, \sqrt[3]{49}. \end{multline*} So, we have $$ \mathcal{S}_{n+3}^{*} = \sqrt[3]{49}\, \mathcal{S}_{n+1}^{*} + \mathcal{S}_{n}^{*}, \qquad n\in \mathbb{N}_0, $$ which implies $$ \mathcal{S}_{n}^{*} = a_{n}^{*}+ b_{n}^{*}\, \sqrt[3]{7} + c_{n}^{*}\, \sqrt[3]{49}, $$ where \begin{align*} {a}^{*}_{n+3} &= {a}^{*}_n + 7\, {b}^{*}_{n+1},\\ {b}^{*}_{n+3} &= {b}^{*}_n + 7\, {c}^{*}_{n+1},\\ {c}^{*}_{n+3} &= {c}^{*}_n + {a}^{*}_{n+1},%\\ \end{align*} \begin{align*} {a}^{*}_0 &=3, \quad {a}^{*}_1=0,\quad {a}^{*}_2=0,\\ {b}^{*}_0 &=0, \quad {b}^{*}_1=0,\quad {b}^{*}_2=0,\\ {c}^{*}_0 &=0, \quad {c}^{*}_1=0,\quad {c}^{*}_2=2. \end{align*} We note that the elements $\mathcal{S}_{3n}=\widehat{a}_{n}$ and $\mathcal{S}_{3n}^{*}={a}^{*}_{n}$, $n=0,1,\ldots$, are all integers. \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Some theoretical deliberations}\label{roz3a} Let us assume that $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ are complex roots of the following polynomial with complex coefficients $$ f(z) := z^{3} + p\,z^{2} + q\,z + r. $$ The symbols $\sqrt[3]{\xi_{1}}$, $\sqrt[3]{\xi_{2}}$, $\sqrt[3]{\xi_{3}}$ will denote any of the third complex roots of the numbers $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$, respectively (only in the case where $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$ are real numbers, we will assume that $\sqrt[3]{\xi_{1}}$, $\sqrt[3]{\xi_{2}}$ and $\sqrt[3]{\xi_{3}}$ also denote the respective real roots). Let us assume that $$ A := \left( \sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}} \right)^{3} $$ and $$ B := \left( \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} + \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \right)^{3}. $$ Thus, the numbers $$ \sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}} \qquad \mbox{ and } \qquad \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} + \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} $$ belong to the sets of the third complex roots of $A$ and~$B$, respectively, which, for conciseness of notation, will be denoted by the symbols $\sqrt[3]{A}$ and~$\sqrt[3]{B}$, respectively. In other words, we have $$ \sqrt[3]{\xi_{1}} + \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{3}} \in \sqrt[3]{A} $$ and $$ \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}} + \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{3}} + \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \in \sqrt[3]{B} . $$ Hence, after two-sided raising of the numbers to the third power, we obtain the following formulas: $$ A = \xi_{1} + \xi_{2} + \xi_{3} + 3\, \sum _{k \neq l} \left( \sqrt[3]{\xi_{k}} \right)^{2}\, \sqrt[3]{\xi_{l}} + 6\, \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} , $$ and $$ B = \xi_{1}\, \xi_{2} + \xi_{1}\, \xi_{3} + \xi_{2}\, \xi_{3} + 3\! \sum _{k \neq l} \left( \sqrt[3]{\xi_{k}} \right)^{2}\! \sqrt[3]{\xi_{l}}\, \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3} }+ 6\! \left( \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \right)^{2}, $$ where $$ \sqrt[3]{\xi_{1}}\, \sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}} \in \sqrt[3]{\xi_{1}\, \xi_{2} \,\xi_{3}} = \sqrt[3]{-r} = - \sqrt[3]{r}. $$ Also here, for abbreviation, the product $({-}1)\, \sqrt[3]{\xi_{1}}\,\sqrt[3]{\xi_{2}}\, \sqrt[3]{\xi_{3}}$ will be denoted by the symbol $\sqrt[3]{r}$. Taking into account Vi\`ete's formulas for polynomial $f(z)$, the expressions for $A$ and~$B$ can be attributed the following form (from now on, symbols $\sqrt[3]{A}$ and~$\sqrt[3]{B}$ will mean the properly selected elements from sets $\sqrt[3]{A}$ and~$\sqrt[3]{B}$, respectively): \begin{equation}\label{ram-w1} A = -p + 3\, \sqrt[3]{A}\, \sqrt[3]{B} + 3\sqrt[3]{r} \end{equation} and \begin{equation}\label{ram-w2} B = q - 3\, \sqrt[3]{A}\, \sqrt[3]{B}\, \sqrt[3]{r} - 3\, \left( \sqrt[3]{r} \right) ^{2} . \end{equation} By multiplying the first of these equations by $\sqrt[3]{r}$ and adding the equations side-by-side, we obtain \begin{equation}\label{ram-w3} B = q - (A + p)\, \sqrt[3]{r} . \end{equation} At the same time, the equation~(\ref{ram-w1}) yields $$ 3\, \sqrt[3]{A}\, \sqrt[3]{B} = A + p - 3\, \sqrt[3]{r} , $$ i.e., $$ 27\, A\, B = \left( A + p - 3\, \sqrt[3]{r} \right) ^{3} , $$ hence, with respect to~(\ref{ram-w3}) we obtain \begin{multline*} 27\, A\, \big( q - (A + p)\, \sqrt[3]{r} \big) = A^{3} + p^{3} - 27\, r + {}\\ {}+ 3\, \big( A^{2}\, (p - 3\sqrt[3]{r}) + A\, (p^{2} + 9\, (\sqrt[3]{r})^{2} ) - 3\, p^{2}\, \sqrt[3]{r} + 9\, p\, (\sqrt[3]{r})^{2} \big) - 18\, A\, p\, \sqrt[3]{r}, \end{multline*} and having rearrange the summands (with respect to the powers of~$A$), we obtain the basic equality \begin{equation}\label{ram-w4} A^{3} + 3\, \big( p + 6\, \sqrt[3]{r} \big) A^{2} + 3\, \big( p^{2} + 3\, p\, \sqrt[3]{r} + 9\, (\sqrt[3]{r})^{2} - 9\, q \big) A + \big( p - 3\, \sqrt[3]{r} \big)^{3} = 0. \end{equation} By applying Cardano's formula to this polynomial, we get the following basic formula (the right side of the formula below means a~properly selected third root of the number present in the formula): \begin{multline}\label{ram-gw} \sqrt[3]{A}=\sqrt[3]{\xi_1} +\sqrt[3]{\xi_2}+\sqrt[3]{\xi_3} = {}\\ {}= \sqrt[3]{ -p-6\, \sqrt[3]{r} - \frac{3}{\sqrt[3]{2}}\, \Big( \sqrt[3]{\mathcal{S}+\sqrt{\mathcal{T}}} + \sqrt[3]{\mathcal{S}-\sqrt{\mathcal{T}}}\, \Big) }, \end{multline} where \begin{align*} \mathcal{S} &:= p\, q +6\, q\, \sqrt[3]{r} +6\, p\, \sqrt[3]{r^2} +9\, r,\\ \mathcal{T} &:= p^2\, q^2 -4\, q^3 - 4\, p^3\, r + 18\, p\,q\, r -27\, r^2. \end{align*} In the case when $\mathcal{T}\geq 0$, $\mathcal{S}\in \mathbb{R}$, $r\in \mathbb{R}$, we assume that all the roots appearing here are real. \begin{remark}\label{ram-rem1} We note that, if in the formula~(\ref{ram-w4}) the following condition holds $$ \big( p + 6\, \sqrt[3]{r} \big)^{2} = p^{2} + 3\, p\, \sqrt[3]{r} + 9\, (\sqrt[3]{r})^{2} - 9\, q , $$ i.e., \begin{equation}\label{ram-w5} p\, \sqrt[3]{r} + 3\, (\sqrt[3]{r})^{2} + q = 0, \end{equation} then the equation~(\ref{ram-w4}) could be given in the form $$ \big( A + p + 6\, \sqrt[3]{r} \big)^{3} = \big( p + 6\, \sqrt[3]{r} \big)^{3} - \big( p - 3\, \sqrt[3]{r} \big) ^{3} , $$ hence we get \begin{equation}\label{ram-w6} A = - p - 6\, \sqrt[3]{r} + \sqrt[3]{ \big( p + 6\, \sqrt[3]{r} \big)^{3} - \big( p - 3\, \sqrt[3]{r} \big)^{3} }. \end{equation} \end{remark} \begin{remark} The analysis which enabled describing the value of~$A$ by means of coefficients of polynomial~$f(z)$ comes from the papers~\cite{Shevelev1988,Kreczmar} (see also~\cite{GrzymkowskiWitula}). \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Basic sequences}\label{roz3} We will now provide definitions of a~dozen basic sequences (not only integer sequences) used further in the paper. For more information concerning these sequences (including the trigonometric relationships defining their terms), see the papers~\cite{WitulaSlota-A7,WitulaSlota-A7s}. The sequences $\{A_{n}(\delta)\}_{n=0}^{\infty}$, $\{B_{n}(\delta)\}_{n=0}^{\infty}$ and $\{C_{n}(\delta)\}_{n=0}^{\infty}$ are the so-called quasi-Fibonacci numbers of the seventh order described in~\cite{WitulaSlota-A7} by means of 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 the seventh order ($\xi^7=1$ and $\xi\neq 1$), $\delta\in \mathbb{C}$, $\delta\neq 0$. These sequences satisfy the following recurrence relations \begin{equation} \left\{\begin{array}{l} A_0(\delta )=1,\ \ B_0(\delta )=C_0(\delta )=0, \\ A_{n+1}(\delta )=A_n(\delta )+2\,\delta\, B_n(\delta )-\delta\, C_n(\delta ), \\ B_{n+1}(\delta )=\delta\, A_n(\delta )+B_n(\delta ), \\ C_{n+1}(\delta )=\delta\, B_n(\delta )+(1-\delta )\,C_n(\delta ), \end{array}\right. \label{aa} \end{equation} for every $n\in\mathbb{N}$. Two auxiliary sequences $\{\mathcal{A}_{n}(\delta)\}_{n=0}^{\infty}$ and $\{\mathcal{B}_{n}(\delta)\}_{n=0}^{\infty}$ connected with these ones are defined by the following relations: \begin{equation}\label{w-and} \mathcal{A}_{n}(\delta) := 3\, A_{n}(\delta) - B_{n}(\delta) -C_{n}(\delta) \end{equation} and \begin{equation} \mathcal{B}_{n}(\delta) := \frac{1}{2} \big( \big(\mathcal{A}_{n}(\delta)\big)^{2} - \mathcal{A}_{2n}(\delta) \big).\label{w-bnd} \end{equation} Furthermore, to simplify notation, we will write \begin{equation}\label{nr100} \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 elements of the sequences $\{\mathcal{A}_{n}\}_{n=0}^{\infty}$, $\{{A}_{n}\}_{n=0}^{\infty}$, $\{{B}_{n}\}_{n=0}^{\infty}$ and $\{{C}_{n}\}_{n=0}^{\infty}$ respectively, satisfy the following recurrence relation (see~\cite[eq.~(3.20)]{WitulaSlota-A7s}): $$ \mathbb{X}_{n+3} -2\, \mathbb{X}_{n+2} -\mathbb{X}_{n+1} + \mathbb{X}_n \equiv 0. $$ Simultaneously, the elements of sequence $\{\mathcal{B}_{n}\}_{n=0}^{\infty}$ by~\cite[eqs.~(3.18), (3.13)]{WitulaSlota-A7s} satisfy the following relation \begin{equation}\label{w-Nr-0} \mathcal{B}_{n+3} + \mathcal{B}_{n+2} -2\, \mathcal{B}_{n+1} - \mathcal{B}_n \equiv 0. \end{equation} \begin{remark} The sequence $\{\mathcal{B}_{n}\}_{n=0}^{\infty}$ is an accelerator sequence for Catalan's constant (see~\cite[sequence \seqnum{A094648}]{Sloane} and papers~\cite{WitulaSlota-A7,WitulaSlota-A7s}). \end{remark} The elements of the sequences $\{a_n\}_{n=0}^{\infty}$, $\{b_n\}_{n=0}^{\infty}$ and $\{c_n\}_{n=0}^{\infty}$ are defined by the following recurrence relations: $$ a_{0}=b_{0}=c_{0}=\sqrt{7} $$ and \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 $n=0,1,2,\ldots$. Moreover, we will use the following sequences \begin{equation}\label{f2s-w2ga} \left\{ \begin{array}{l} \overline{\alpha}_{n} := c_{n+1},\\ \overline{\beta}_{n} := {-}a_{n} - b_{n} ,\\ \overline{\gamma}_{n} := a_{n},\\ \end{array} \right. \end{equation} Next, sequences $\{f_n\}_{n=0}^{\infty}$, $\{g_n\}_{n=0}^{\infty}$, $\{h_n\}_{n=0}^{\infty}$ and $\{H_n\}_{n=0}^{\infty}$ are defined in the following way $$ f_0=g_0=h_0=-1, $$ 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} = \mathcal{B}_{n+1},& \quad n \geq 1,\\ 2\, H_n = f_{n}^{2} +f_{2n} - g_{n}^{2} + h_{n}^{2} - h_{2n+1} -2\, h_{2n}, & \quad n \geq 0,\\ \end{array} \right. \end{equation} (the numbers $ \mathcal{B}_{n}$ are defined by the formula~(\ref{nr100}) above). And at last, the elements of sequences $\{u_n\}_{n=0}^{\infty}$, $\{v_n\}_{n=0}^{\infty}$, $\{w_n\}_{n=0}^{\infty}$, $\{x_n\}_{n=0}^{\infty}$, $\{y_n\}_{n=0}^{\infty}$ and $\{z_n\}_{n=0}^{\infty}$ are defined by \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 $n=0,1,2,\ldots$, where $u_0=v_0=w_0=-1$, $x_0=y_0=z_0=\sqrt{7}$ and $z_1=7$. \begin{remark} All the recurrence sequences above have a~third order; the selective identities, Binet formulas and generating functions, and some different identities for these numbers, are presented in papers \cite{WitulaSlota-A7,WitulaSlota-A7s}. For example, the following equivalent recurrence relations hold $$ \mbox{(\ref{f2s-w1g})}\ \ \Longleftrightarrow\ \ \mathbb{X}_{n+3} - 5\, \mathbb{X}_{n+2} + 6\, \mathbb{X}_{n+1} - \mathbb{X}_{n} = 0, $$ for $n=0,1,2,\ldots$, and $\mathbb{X}\in \{a,b,c\}$, and $a_{0}=b_{0}=c_{0}=\sqrt{7}$, $a_{1}=3\,\sqrt{7}$, $b_{1}=2\,\sqrt{7}$, $c_{1}=0$, $a_{2}=8\,\sqrt{7}$, $b_{2}=\sqrt{7^3}$, $c_{1}=-2\, \sqrt{7}$. We also have \begin{align*} a_{n} &= 2^{2n+1}\, \Big[ \sin \alpha\, \big(\! \cos 4\alpha \big)^{\!2n} + \sin 2\alpha\, \big(\! \cos \alpha \big)^{\!2n} + \sin 4\alpha\, \big(\! \cos 2\alpha \big)^{\!2n} \Big],\\ b_{n} &= 2^{2n+1}\, \Big[ \sin 2\alpha\, \big(\! \cos 4\alpha \big)^{\!2n} + \sin 4\alpha\, \big(\! \cos \alpha \big)^{\!2n} + \sin \alpha\, \big(\! \cos 2\alpha \big)^{\!2n} \Big],\\ c_{n} &= 2^{2n+1}\, \Big[ \sin 4\alpha\, \big(\! \cos 4\alpha \big)^{\!2n} + \sin \alpha\, \big(\! \cos \alpha \big)^{\!2n} + \sin 2\alpha\, \big(\! \cos 2\alpha \big)^{\!2n} \Big], \end{align*} etc. \end{remark} \begin{remark} Now we present some new identities for the above sequences (identities~(\ref{nr-alpha})--(\ref{nr-delta}), (\ref{w3.16})--(\ref{w3.19})), which will be used in subsection~\ref{poroz4.2}. These identities significantly complete those obtained in paper~\cite{WitulaSlota-A7s}. By~\cite[Lemma~3.14 (a)]{WitulaSlota-A7s}, equality~(\ref{w-0}) and \cite[eq.~(3.21)]{WitulaSlota-A7s}, the following identity holds \begin{equation}\label{nr-alpha} (-1)^{n}\, \mathcal{A}_{n} = H_{n+1}. \end{equation} Hence, by~\cite[eq.~(3.23)]{WitulaSlota-A7s}, we obtain \begin{multline}\label{nr-beta} 2\, \cos \alpha\, \big( 2\, \cos 2\alpha \big)^{-n} + 2\, \cos 2\alpha\, \big( 2\, \cos 4\alpha \big)^{-n} + 2\, \cos 4\alpha\, \big( 2\, \cos \alpha \big)^{-n} ={}\\ {}= (-1)^{n}\, \big( \mathcal{A}_{n} + \mathcal{A}_{n-1} - \mathcal{A}_{n-2} \big) {}= (-1)^{n}\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n} \big), \end{multline} and next, by~\cite[eq.~(3.22)]{WitulaSlota-A7s}, we obtain \begin{multline}\label{nr-gamma} 2\, \cos \alpha\, \big( 2\, \cos 4\alpha \big)^{-n} + 2\, \cos 2\alpha\, \big( 2\, \cos \alpha \big)^{-n} + 2\, \cos 4\alpha\, \big( 2\, \cos 2\alpha \big)^{-n} ={}\\ {}= (-1)^{n}\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1} + \mathcal{A}_{n-2} - 7\, A_{n} \big) {}= (-1)^{n}\, \big( \mathcal{A}_{n-1} - \mathcal{A}_{n+1} \big), \end{multline} since, by~\cite[Remark~3.11]{WitulaSlota-A7s}, we have the identity $$ \mathcal{A}_{n+3} = 2\, \mathcal{A}_{n+2} + \mathcal{A}_{n+1} - \mathcal{A}_{n} $$ and by~\cite[Remark~3.8]{WitulaSlota-A7s}, we have $$ 4\, \mathcal{A}_{n} - \mathcal{A}_{n-2} = 7\, {A}_{n}. $$ Moreover, the following formula can be easily generated \begin{equation}\label{nr-delta} \mathcal{B}_{n} = g_{n+1} + h_{n+1}. \end{equation} By~\cite[eq.~(3.18)]{WitulaSlota-A7s}, we have \begin{equation}\label{w3.15} h_{n-1}=\mathcal{B}_{n}, \end{equation} which by~(\ref{nr-delta}) implies \begin{equation}\label{w3.16} g_{n+1}=\mathcal{B}_{n}-\mathcal{B}_{n+2}, \end{equation} and next, by~\cite[eq.~(3.12)]{WitulaSlota-A7s} and by~(\ref{w-Nr-0}), we get \begin{equation}\label{w3.17} f_{n}=g_{n+1}-h_{n}=\mathcal{B}_{n}-\mathcal{B}_{n+2}-\mathcal{B}_{n+1}= -\mathcal{B}_{n-1}-\mathcal{B}_{n}. \end{equation} Moreover, we obtain \begin{equation}\label{w3.18} f_{n}+g_{n}=-\mathcal{B}_{n}-\mathcal{B}_{n+1} \end{equation} and \begin{equation}\label{w3.19} f_{n}+h_{n}=\mathcal{B}_{n}-\mathcal{B}_{n+2}. \end{equation} \end{remark} \begin{remark}\label{rem4-4} From~(\ref{w3.15}) we get (see~\cite[eq.~(3.11)]{WitulaSlota-A7s}): $$ \mathcal{B}_{n} = 2^n\, \big( \cos^n\alpha + \cos^n 2\alpha + \cos^n 4\alpha \big), $$ from which the next form can be deduced \begin{equation}\label{nnnnn} \mathcal{B}_{n} = \Big( \frac{\sin 2\alpha}{\sin \alpha} \Big)^{\! n} + \Big( \frac{\sin 4\alpha}{\sin 2\alpha} \Big)^{\! n} + \Big( \frac{\sin \alpha}{\sin 4\alpha} \Big)^{\! n}. \end{equation} This forms of Binet's formula for $\mathcal{B}_{n}$ are more attractive than the Sloane's ones (see~\cite[sequence \seqnum{A094648}]{Sloane}). Moreover, the formula~(\ref{nnnnn}) makes it possible to extend the definition $\mathcal{B}_{n}$ for negative integers~$n$. So $\{\mathcal{B}_{n}\}_{n=-\infty}^{\infty}$ is a~two-sided sequence of integers which is defined for all integers either by recurrence formula~(\ref{w-Nr-0}), or equivalently by Binet's formula~(\ref{nnnnn}). \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Applications of the formula~(\ref{ram-gw})} \subsection{Some special formulas} \subsubsection*{By~\cite[eq.~(3.30)]{WitulaSlota-A7s}} \noindent for $n=1$: \begin{equation}\label{sf-w1} \sqrt[3]{\sec \alpha} + \sqrt[3]{\sec 2\alpha} + \sqrt[3]{\sec 4\alpha} = \sqrt[3]{8-6\,\sqrt[3]{7}}\, ; \end{equation} %\medskip \noindent for $n=2$: \begin{equation}\label{sf-w2} \sqrt[3]{\frac{\cos \alpha}{\cos 2\alpha}}+ \sqrt[3]{\frac{\cos 2\alpha}{\cos 4\alpha}}+ \sqrt[3]{\frac{\cos 4\alpha}{\cos \alpha}}= -\sqrt[3]{7}\, ; \end{equation} %\medskip \noindent for $n=3$: \begin{equation}\label{sf-w3} \cos 2\alpha\, \sqrt[3]{2\, \cos \alpha} + \cos 4\alpha\, \sqrt[3]{2\, \cos 2\alpha} + \cos \alpha\, \sqrt[3]{2\, \cos 4\alpha} %={}\\ {}= \frac{1}{2}\, \sqrt[3]{5 + 3\, \sqrt[3]{7} - 3\, \sqrt[3]{49}} \, ; \end{equation} %\medskip \noindent for $n=4$: \begin{equation}\label{sf-w4} \cos 2\alpha\, \sqrt[3]{\sec 4\alpha} + \cos 4\alpha\, \sqrt[3]{\sec \alpha} + \cos \alpha\, \sqrt[3]{\sec 2\alpha} = -\sqrt[3]{\frac{3}{4}\, \big( 1 + \sqrt[3]{7} \big)^{2} } \, ; \end{equation} %\medskip \noindent for $n=5$: \begin{equation}\label{sf-w5} \cos 2\alpha\, \sqrt[3]{ \frac{\cos 2\alpha}{ \cos 4\alpha} } + \cos 4\alpha\, \sqrt[3]{ \frac{\cos 4\alpha}{ \cos \alpha} } + \cos \alpha\, \sqrt[3]{ \frac{\cos \alpha}{ \cos 2\alpha} } = 0 \, ; \end{equation} %\medskip \noindent for $n=6$: \begin{multline}\label{sf-w6} \cos^2 (2\alpha) \, \sqrt[3]{2\, \cos \alpha} + \cos^2 (4\alpha) \, \sqrt[3]{2\, \cos 2\alpha} + \cos^2 (\alpha) \, \sqrt[3]{2\, \cos 4\alpha} ={}\\ {}= \frac{1}{4}\, \sqrt[3]{12\, \sqrt[3]{7} - \big( 4 + 3\, \sqrt[3]{7}\,\big)^2 } \, ; \end{multline} %\medskip \noindent for $n=7$: \begin{equation}\label{sf-w7} \cos^2 (2\alpha) \, \sqrt[3]{\sec 4\alpha} + \cos^2 (4\alpha) \, \sqrt[3]{\sec \alpha} + \cos^2 (\alpha)\, \sqrt[3]{\sec 2\alpha} %={}\\ {}= \frac{1}{4}\, \sqrt[3]{2\,\big( 5 - 3\, \sqrt[3]{7}\,\big)^2 } \, ; \end{equation} %\medskip \noindent for $n=8$: \begin{multline}\label{sf-w7a} \cos^3 (2\alpha)\, \sqrt[3]{ \frac{\cos \alpha}{ \cos 2\alpha} } + \cos^3 (4\alpha)\, \sqrt[3]{ \frac{\cos 2\alpha}{ \cos 4\alpha} } + \cos^3 (\alpha)\, \sqrt[3]{ \frac{\cos 4\alpha}{ \cos \alpha} } ={}\\ {}= \frac{1}{2}\, \bigg(\, \cos^2 (2\alpha)\, \sqrt[3]{ \frac{\cos 2\alpha}{ \cos 4\alpha} } + \cos^2 (4\alpha)\, \sqrt[3]{ \frac{\cos 4\alpha}{ \cos \alpha} } + \cos^2 (\alpha)\, \sqrt[3]{ \frac{\cos \alpha}{ \cos 2\alpha} }\, \bigg) = -\frac{3}{8}\, \sqrt[3]{7}\, , \end{multline} which also can be generated from ~(\ref{sf-w2}). \medskip \begin{remark} The formula~(\ref{sf-w7}) also follows from~(\ref{w-1.0}) and~(\ref{w-0}) for $\mathbb{X}=0$: \begin{multline*} 2\, \cos^2 (2\alpha) \, \sqrt[3]{\sec 4\alpha} + 2\, \cos^2 (4\alpha) \, \sqrt[3]{\sec \alpha} + 2\, \cos^2 (\alpha)\, \sqrt[3]{\sec 2\alpha} ={}\\ {}= \sum_{k=0}^{2} \sqrt[3]{\sec ( 2^{k}\alpha )} + \sum_{k=0}^{2} \Big( \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2 = {}\\ {}= \sum_{k=0}^{2} \sqrt[3]{\sec ( 2^{k}\alpha )} + \Big( \sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2 -2\, \sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )\, \cos ( 2^{k+1}\alpha )} = {}\\ {}= \Big( \sum_{k=0}^{2} \sqrt[3]{\cos ( 2^{k}\alpha )}\, \Big)^2. \end{multline*} \end{remark} \subsubsection*{By~\cite[eq.~(3.31)]{WitulaSlota-A7s}} \noindent for $n=2$: \begin{equation}\label{sf-w8} \sqrt[3]{ \frac{\cos 2\alpha}{ \cos \alpha} } + \sqrt[3]{ \frac{\cos 4\alpha}{ \cos 2\alpha} } + \sqrt[3]{ \frac{\cos \alpha}{ \cos 4\alpha} } = 0 \, ; \end{equation} %\medskip \noindent for $n=3$: \begin{equation}\label{sf-w9} \cos \alpha\, \sqrt[3]{\cos 2\alpha} + \cos 2\alpha\, \sqrt[3]{\cos 4\alpha} + \cos 4\alpha\, \sqrt[3]{\cos \alpha} = -\frac{1}{2}\, \sqrt[3]{1 + \frac{3}{2}\, \sqrt[3]{49}} \, ; \end{equation} %\medskip \noindent for $n=4$: \begin{equation}\label{sf-w10} \cos \alpha\, \sqrt[3]{\sec 4\alpha} + \cos 4\alpha\, \sqrt[3]{\sec 2\alpha} + \cos 2\alpha\, \sqrt[3]{\sec \alpha} = \sqrt[3]{\frac{9}{4}\, \big( 2 - \sqrt[3]{7} \big) } \, ; \end{equation} %\medskip \noindent for $n=5$: \begin{equation}\label{sf-w11} \cos^2 (2\alpha)\, \sqrt[3]{ \frac{\cos 4\alpha}{ \cos 2\alpha} } + \cos^2 (4\alpha)\, \sqrt[3]{ \frac{\cos \alpha}{ \cos 4\alpha} } + \cos^2 (\alpha)\, \sqrt[3]{ \frac{\cos 2\alpha}{ \cos \alpha} } = -\frac{1}{4}\, \sqrt[3]{49} ; \end{equation} %\medskip \noindent for $n=6$: \begin{multline}\label{sf-w12} \cos^2 (2\alpha) \, \sqrt[3]{\cos 4\alpha} + \cos^2 (\alpha) \, \sqrt[3]{\cos 2\alpha} + \cos^2 (4\alpha) \, \sqrt[3]{\cos \alpha} ={}\\ {}= 2^{-7/3}\, \sqrt[3]{47 + 3 \, \sqrt[3]{7} - 12\, \sqrt[3]{49} } \, ; \end{multline} %\medskip \noindent for $n=7$: \begin{multline}\label{sf-w13} \cos^2 (\alpha) \, \sqrt[3]{\sec 4\alpha} + \cos^2 (4\alpha) \, \sqrt[3]{\sec 2\alpha} + \cos^2 (2\alpha) \, \sqrt[3]{\sec \alpha} ={}\\ {}= -2^{5/3}\, \sqrt[3]{ 73 + 36\, \sqrt[3]{7} + 3\, \sqrt[3]{49} } \, ; \end{multline} %\medskip \begin{remark} We note that \begin{multline*} 2\, \cos^2 (2\alpha) \, \sqrt[3]{2\, \cos \alpha} + 2\, \cos^2 (4\alpha) \, \sqrt[3]{2\, \cos 2\alpha} + 2\, \cos^2 (\alpha) \, \sqrt[3]{2\, \cos 4\alpha} ={}\\ {}= \Big( \sqrt[3]{2\, \cos \alpha}+ \sqrt[3]{2\, \cos 2\alpha}+ \sqrt[3]{2\, \cos 4\alpha} \Big) + {}\\ {}+ \cos 4\alpha \, \sqrt[3]{2\, \cos \alpha} + \cos \alpha \, \sqrt[3]{2\, \cos 2\alpha} + \cos 2\alpha \, \sqrt[3]{2\, \cos 4\alpha} ={}\\ {}\stackrel{(\ref{w-1.0}), (\ref{sf-w9})}{=} \sqrt[3]{5-3\, \sqrt[3]{7}} - \frac{1}{2}\, \sqrt[3]{2+3\, \sqrt[3]{49}} \stackrel{(\ref{sf-w6})}{=} \frac{1}{2}\, \sqrt[3]{12\, \sqrt[3]{7} - \big( 4 + 3\, \sqrt[3]{7}\big)^{2}\, }, \end{multline*} which implies the identity $$ \sqrt[3]{16 + 12\, \sqrt[3]{7} + 9\, \sqrt[3]{49}\, } = 2\, \sqrt[3]{3\, \sqrt[3]{7}-5} + \sqrt[3]{2+3\, \sqrt[3]{49}}. $$ \end{remark} \subsubsection*{By~\cite[eq.~(3.32)]{WitulaSlota-A7s}} \noindent for $n=2$: \begin{equation}\label{sf-w14} \sqrt[3]{ \frac{\cos \alpha}{ \cos^2 (2\alpha)} } + \sqrt[3]{ \frac{\cos 2\alpha}{ \cos^2 (4\alpha)} } + \sqrt[3]{ \frac{\cos 4\alpha}{ \cos^2 (\alpha)} } = \sqrt[3]{ 2\, \big( 11 - 3\, \sqrt[3]{49} \big) } \, ; \end{equation} %\medskip \noindent for $n=3$: \begin{equation}\label{sf-w15} \frac{\sqrt[3]{\cos \alpha}}{ \cos 2\alpha} + \frac{\sqrt[3]{\cos 2\alpha}}{ \cos 4\alpha} + \frac{\sqrt[3]{\cos 4\alpha}}{ \cos \alpha} = -\sqrt[3]{ 36\, \big(1 +\sqrt[3]{7} \big) } \, ; \end{equation} %\medskip \subsubsection*{By~\cite[eq.~(3.33)]{WitulaSlota-A7s}} \noindent for $n=2$: \begin{equation}\label{sf-w16} \sqrt[3]{ \frac{\cos \alpha}{ \cos^2 (4\alpha)} } + \sqrt[3]{ \frac{\cos 2\alpha}{ \cos^2 (\alpha)} } + \sqrt[3]{ \frac{\cos 4\alpha}{ \cos^2 (2\alpha)} } %= {}\\ {}= \sqrt[3]{ 6\, \big(-1 + \sqrt[3]{7} - \sqrt[3]{49}\big) } = -2\, \sqrt[3]{\frac{6}{1+\sqrt[3]{7}}} \, ; \end{equation} %\medskip \noindent for $n=3$: \begin{equation}\label{sf-w17} \frac{\sqrt[3]{\cos \alpha}}{ \cos 4\alpha} + \frac{\sqrt[3]{\cos 2\alpha}}{ \cos \alpha} + \frac{\sqrt[3]{\cos 4\alpha}}{ \cos 2\alpha} = \sqrt[3]{ 4\, \big(26 -6\, \sqrt[3]{7} - 3\, \sqrt[3]{49}\big) } \, ; \end{equation} %\medskip \subsubsection*{By~\cite[eq.~(3.34)]{WitulaSlota-A7s}} \noindent for $n=2$ we get~(\ref{sf-w1}); \medskip \noindent for $n=3$: \begin{equation}\label{sf-w19} \big( 2\, \cos \alpha \big)^{-2/3} + \big( 2\, \cos 2\alpha \big)^{-2/3} + \big( 2\, \cos 4\alpha \big)^{-2/3} %= {}\\ {}= \sqrt[3]{ 12 + 6\, \sqrt[3]{7} + 3\, \sqrt[3]{49} }\, ; \end{equation} %\medskip \subsubsection*{By~\cite[eq.~(4.30)]{WitulaSlota-A7s}} \noindent for $n=1$: \begin{multline}\label{sf-w20} \sqrt[18]{7} \, \Big( \sqrt[3]{\cot \alpha} + \sqrt[3]{\cot 2\alpha} + \sqrt[3]{\cot 4\alpha}\, \Big) ={}\\ = \sqrt[3]{ \sqrt[3]{49} -6 + 3\, \sqrt[3]{3\, (1-\sqrt[3]{7}+\sqrt[3]{49})} -3\, \sqrt[3]{5+3\,\sqrt[3]{7}-3\,\sqrt[3]{49}}\, } \, ; \end{multline} %\medskip \noindent for $n=2$: \begin{multline}\label{sf-w22} \sqrt[9]{\frac{7}{8}} \, \Big( \sqrt[3]{\cot \alpha\, \csc \alpha } + \sqrt[3]{\cot 2\alpha\, \csc 2\alpha} + \sqrt[3]{\cot 4\alpha\, \csc 4\alpha}\, \Big) ={}\\ = \sqrt[3]{6 - 2\, \sqrt[3]{7} - 3\, \sqrt[3]{3\, (1+\sqrt[3]{7})^2} - 3\, \sqrt[3]{-26+6\,\sqrt[3]{7}+3\,\sqrt[3]{49}} } \, , \end{multline} i.e, \begin{equation}\label{sf-w22a} \sum_{k=0}^{2} \sqrt[3]{\cot 2^k\alpha\, \csc 2^k\alpha } %= {}\\ {}= \sqrt[3]{ \frac{6}{\sqrt[3]{7}}\, \Big( 2 - \frac{2}{3}\, \sqrt[3]{7} - \sqrt[3]{3(1+\sqrt[3]{7})^2} - \sqrt[3]{3(1+\sqrt[3]{7})^2-27}\, \Big)} \, . \end{equation} %\medskip \noindent Moreover from~(\ref{gwiazdka}) below we have for $n=1$: \begin{multline}\label{sf-w21} \sqrt[3]{\tan \alpha} + \sqrt[3]{\tan 2\alpha} + \sqrt[3]{\tan 4\alpha} ={}\\ = \sqrt[18]{7} \, \sqrt[3]{ 3\, \sqrt[3]{3\, (1-\sqrt[3]{7}+\sqrt[3]{49})} -3\, \sqrt[3]{5+3\,\sqrt[3]{7}-3\,\sqrt[3]{49}} - 6 - \sqrt[3]{7} } \, ; \end{multline} %\medskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% SubSection %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Some general formulas}\label{poroz4.2} \textbf{1.} By~\cite[eq.~(2.1)]{WitulaSlota-A7s} we obtain (for $n=0,1,2,\ldots$): \begin{multline}\label{ram-2-w1} \big( 2\, \cos \alpha \big)^{n/3} + \big( 2\, \cos 2\alpha \big)^{n/3} + \big( 2\, \cos 4\alpha \big)^{n/3} = {}\\ {}= \sqrt[3]{ \mathcal{B}_{n} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,1} + \sqrt{\mathcal{T}_{n,1}} } + \sqrt[3]{ \mathcal{S}_{n,1} - \sqrt{\mathcal{T}_{n,1}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,1} &:= (-1)^{n-1}\, \mathcal{A}_{n}\, \big( \mathcal{B}_{n} + 6 \big) - 6\,\mathcal{B}_{n} -9,\label{ram-2-w2}\\ \mathcal{T}_{n,1} &:= \mathcal{A}_{n}^{2}\,\mathcal{B}_{n}^{2} + 4\, (-1)^{n-1}\, \mathcal{A}_{n}^{3} - 4\,\mathcal{B}_{n}^{3} + 18\, (-1)^{n}\, \mathcal{A}_{n}\,\mathcal{B}_{n} -27.\label{ram-2-w3} \end{align} We note that \begin{equation}\label{ram-2-w4} \mathcal{B}_{n} = \big( 2\, \cos \alpha \big)^{n} + \big( 2\, \cos 2\alpha \big)^{n} + \big( 2\, \cos 4\alpha \big)^{n}, \end{equation} so we get the following interesting identity \begin{equation}\label{ram-2-w5} \frac{1}{3}\, \Big(6 + \mathcal{B}_{3n} - \mathcal{B}_{n}^{3}\Big) = \sqrt[3]{ \tfrac{1}{2} \big(\mathcal{S}_{3n,1} + \sqrt{\mathcal{T}_{3n,1}}\,\big) } + \sqrt[3]{ \tfrac{1}{2} \big(\mathcal{S}_{3n,1} - \sqrt{\mathcal{T}_{3n,1}}\,\big) }. \end{equation} Moreover, if $n\in \mathbb{N}$ then by~\cite[eq.~(3.34)]{WitulaSlota-A7s} and by~(\ref{nr-alpha}) and~(\ref{nr-delta}) we obtain \begin{multline}\label{ram-2-w1-nw} \big( 2\, \cos \alpha \big)^{-n/3} + \big( 2\, \cos 2\alpha \big)^{-n/3} + \big( 2\, \cos 4\alpha \big)^{-n/3} = {}\\ {}= \sqrt[3]{ (-1)^{n}\, \mathcal{A}_{n} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,1} + \sqrt{\mathcal{T}_{n,1}} } + \sqrt[3]{ \mathcal{S}_{n,1} - \sqrt{\mathcal{T}_{n,1}} }\,\, \Big) }. \end{multline} Hence, for example for $n=2$ the formula~(\ref{sf-w1}) follows. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{2.} By~\cite[eq.~(3.27)]{WitulaSlota-A7s} we obtain \begin{multline}\label{w2-d} \sqrt[3]{ 2\, \sin \alpha\, \big(2\, \cos 4\alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 2\alpha\, \big(2\, \cos \alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 4\alpha\, \big(2\, \cos 2\alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ p_{n,2} - 6 \, \sqrt[6]{7} - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,2} + \sqrt{\mathcal{T}_{n,2}} } + \sqrt[3]{ \mathcal{S}_{n,2} - \sqrt{\mathcal{T}_{n,2}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,2} &= 7\,(-1)^{n}\, {B}_{n}\, \big( 6\, \sqrt[6]{7} - p_{n,2} \big) -6\, \sqrt[3]{7} \, p_{n,2} +9 \, \sqrt{7},\label{w2-e}\\ \mathcal{T}_{n,2} &= 49\, B_{n}^{2} \, \big( p_{n,2}^{2} - 28\, (-1)^n\, B_{n} \big) +{} \nonumber\\ &\phantom{==} +2\, \sqrt{7}\, p_{n,2}\, \big( 2\, p_{n,2}^{2} - 63\, (-1)^n B_{n} \big) -189, \label{w2-f}\\ %\end{align} %\begin{align} p_{n,2}&= \left\{ \begin{array}{ll} a_{n/2} & \mbox{ if $n$ is even},\\ \overline{\alpha}_{(n-1)/2} & \mbox{ if $n$ is odd}, \end{array} \right.\label{w2-a}\\ q_{n,2}&= 7\, (-1)^n\, {B}_{n},\label{w2-b}\\ r_{n,2} &\equiv \sqrt{7}.\label{w2-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{3.} By~\cite[eq.~(3.28)]{WitulaSlota-A7s} we obtain \begin{multline}\label{w3-d} \sqrt[3]{ 2\, \sin \alpha\, \big(2\, \cos 2\alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 2\alpha\, \big(2\, \cos 4\alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 4\alpha\, \big(2\, \cos \alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ p_{n,3} - 6 \, \sqrt[6]{7} - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,3} + \sqrt{\mathcal{T}_{n,3}} } + \sqrt[3]{ \mathcal{S}_{n,3} - \sqrt{\mathcal{T}_{n,3}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,3} &= 7\,(-1)^{n}\, \big( {B}_{n} -{C}_{n} \big)\, \big( p_{n,3} - 6\, \sqrt[6]{7} \big) -6\, \sqrt[3]{7} \, p_{n,3} +9\, \sqrt{7},\label{w3-e}\\ \mathcal{T}_{n,3} &= 49\, \big( {B}_{n} -{C}_{n} \big)^{2} \, \big( p_{n,3}^{2} + 28\, (-1)^n\, \big( {B}_{n} -{C}_{n} \big) \big) +{} \nonumber\\ &\phantom{==} +2\, \sqrt{7}\, p_{n,3}\, \big( 2\, p_{n,3}^{2} + 63\, (-1)^n \big( {B}_{n} -{C}_{n} \big) \big) - 189, \label{w3-f}\\ %\end{align} %\begin{align} p_{n,3}&= \left\{ \begin{array}{ll} b_{n/2} & \mbox{ if $n$ is even},\\ \overline{\beta}_{(n-1)/2} & \mbox{ if $n$ is odd}, \end{array} \right.\label{w3-a}\\ q_{n,3}&= 7\, (-1)^{n-1}\, \big( {B}_{n} -{C}_{n} \big),\label{w3-b}\\ r_{n,3} &\equiv \sqrt{7}.\label{w3-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{4.} By~\cite[eq.~(3.29)]{WitulaSlota-A7s} we obtain \begin{multline}\label{w4-d} \sqrt[3]{ 2\, \sin \alpha\, \big(2\, \cos \alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 2\alpha\, \big(2\, \cos 2\alpha \big)^{n} } + \sqrt[3]{ 2\, \sin 4\alpha\, \big(2\, \cos 4\alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ p_{n,4} - 6 \, \sqrt[6]{7} - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,4} + \sqrt{\mathcal{T}_{n,4}} } + \sqrt[3]{ \mathcal{S}_{n,4} - \sqrt{\mathcal{T}_{n,4}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,4} &= 7\,(-1)^{n}\, {C}_{n}\, \big( p_{n,4} - 6\, \sqrt[6]{7} \big) - 6\, \sqrt[3]{7} p_{n,4} + 9\, \sqrt{7} ,\label{w4-e}\\ \mathcal{T}_{n,4} &= \big( 7\, p_{n,4}\, {C}_{n} + 9\, (-1)^{n}\, \sqrt{7} \big)^{2} \, +4\, \big( 7^3\, (-1)^{n}\, C_{n}^{3} + \sqrt{7}\, p_{n,4}^{3} \big) -756, \label{w4-f}\\ %\end{align} %\begin{align} p_{n,4}&= \left\{ \begin{array}{ll} c_{n/2} & \mbox{ if $n$ is even},\\ \overline{\gamma}_{(n-1)/2} & \mbox{ if $n$ is odd}, \end{array} \right.\label{w4-a}\\ q_{n,4}&= 7\, (-1)^{n-1}\, {C}_{n},\label{w4-b}\\ r_{n,4} &\equiv \sqrt{7}.\label{w4-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{5.} By~\cite[eq.~(3.31)]{WitulaSlota-A7s} we obtain \begin{multline}\label{w5-d} \sqrt[3]{ 2\, \cos \alpha\, \big( 2\, \cos 4\alpha \big)^{n} } + \sqrt[3]{ 2\, \cos 2\alpha\, \big( 2\, \cos \alpha \big)^{n} } + \sqrt[3]{ 2\, \cos 4\alpha\, \big( 2\, \cos 2\alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ g_{n} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,5} + \sqrt{\mathcal{T}_{n,5}} } + \sqrt[3]{ \mathcal{S}_{n,5} - \sqrt{\mathcal{T}_{n,5}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,5} &= -g_{n}\,q_{n,5} - 6 \, \big( g_{n} + q_{n,5} \big)-9,\label{w5-e}\\ \mathcal{T}_{n,5} &= \big( g_{n}\, q_{n,5} +9 \big)^{2} -4\, \big( g_{n}^{3} + q_{n,5}^{3} \big) -108, \label{w5-f} \end{align} \begin{align} p_{n,5}&=-g_{n},\label{w5-a}\\ q_{n,5}&= (-1)^{n}\, \big( \mathcal{A}_{n} + \mathcal{A}_{n+1} - 7\, {A}_{n}\big),\label{w5-b}\\ r_{n,5} &\equiv -1.\label{w5-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{6.} By~\cite[eq.~(3.32)]{WitulaSlota-A7s} and~(\ref{nr-beta}) we obtain \begin{multline}\label{w11-d} \sqrt[3]{ \cos \alpha\, \big( \sec 2\alpha \big)^{n} } + \sqrt[3]{ \cos 2\alpha\, \big( \sec 4\alpha \big)^{n} } + \sqrt[3]{ \cos 4\alpha\, \big( \sec \alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ 2^{n-1}\, \Big( {-}p_{n,6} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,6} + \sqrt{\mathcal{T}_{n,6}} } + \sqrt[3]{ \mathcal{S}_{n,6} - \sqrt{\mathcal{T}_{n,6}} }\,\, \Big) \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,6} &= (-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n}\big)\, \big( \mathcal{B}_{n+2} - \mathcal{B}_{n} - 6 \big) + 6\,\big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big)-9,\label{w11-e}\\ \mathcal{T}_{n,6} &= \big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big)^2\, \Big( \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)^2 + 4\, \big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big) \Big) +{}\nonumber\\ &\phantom{=}+ 2\, (-1)^{n}\,\big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)\, \Big( 2\, \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big)^2 + %{}\\ %&\phantom{=}+ 9\, \big( \mathcal{B}_{n+2} - \mathcal{B}_{n}\big) \Big) - 27,\label{w11-f}\\ %\end{align} %\begin{align} p_{n,6}&=(-1)^n\, \big( \mathcal{A}_{n} - \mathcal{A}_{n+1}\big),\label{w11-a}\\ q_{n,6}&= f_n + h_n = \mathcal{B}_{n} - \mathcal{B}_{n+2},\label{w11-b}\\ r_{n,6} &\equiv -1.\label{w11-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{7.} By~\cite[eq.~(3.33)]{WitulaSlota-A7s} and~(\ref{nr-gamma}) we obtain \begin{multline}\label{w10-d} \sqrt[3]{ \cos \alpha\, \big( \sec 4\alpha \big)^{n} } + \sqrt[3]{ \cos 2\alpha\, \big( \sec \alpha \big)^{n} } + \sqrt[3]{ \cos 4\alpha\, \big( \sec 2\alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ 2^{n-1}\, \Big( {-}p_{n,7} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,7} + \sqrt{\mathcal{T}_{n,7}} } + \sqrt[3]{ \mathcal{S}_{n,7} - \sqrt{\mathcal{T}_{n,7}} }\,\, \Big) \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,7} &= (-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)\, \big( 6 - \mathcal{B}_{n+1} - \mathcal{B}_{n} \big) + 6\,\big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big)-9,\label{w10-e}\\ \mathcal{T}_{n,7} &= \big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big)^2\, \Big( \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)^2 + 4\, \big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big) \Big) +{}\nonumber\\ &+ 2\, (-1)^{n}\,\big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)\, \Big( 2\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big)^2 + 9\, \big( \mathcal{B}_{n+1} + \mathcal{B}_{n}\big) \Big) - 27,\label{w10-f}\\ %\end{align} %\begin{align} p_{n,7}&=(-1)^n\, \big( \mathcal{A}_{n+1} - \mathcal{A}_{n-1}\big),\label{w10-a}\\ q_{n,7}&= f_n + g_n = - \mathcal{B}_{n} - \mathcal{B}_{n+1},\label{w10-b}\\ r_{n,7} &\equiv -1.\label{w10-c} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{8.} By~\cite[eq.~(3.30)]{WitulaSlota-A7s}, we obtain \begin{multline}\label{w6-d} \sqrt[3]{ 2\, \cos \alpha\, \big( 2\, \cos 2\alpha \big)^{n} } + \sqrt[3]{ 2\, \cos 2\alpha\, \big( 2\, \cos 4\alpha \big)^{n} } + \sqrt[3]{ 2\, \cos 4\alpha\, \big( 2\, \cos \alpha \big)^{n} } = {}\\ {}= \sqrt[3]{ f_{n} + 6 - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,8} + \sqrt{\mathcal{T}_{n,8}} } + \sqrt[3]{ \mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,8} &= -f_{n}\,q_{n,8} - 6 \, \big( f_{n} + q_{n,8} \big)-9,\label{w6-e}\\ \mathcal{T}_{n,8} &= \big( f_{n}\, q_{n,8} +9 \big)^{2} -4\, \big( f_{n}^{3} + q_{n,8}^{3} \big) -108, \label{w6-f}\\ %\end{align} %\begin{align} p_{n,8}&=-f_{n},\label{w6-a}\\ q_{n,8}&= (-1)^{n}\, \big( 7\, {A}_{n} - 3\, \mathcal{A}_{n} \big) =(-1)^n \big( \mathcal{A}_{n} - \mathcal{A}_{n-2} \big),\label{w6-b}\\ r_{n,8} &\equiv -1.\label{w6-c} \end{align} \begin{remark}\label{rem4.3} As results from direct observation of the value of expression $$ \sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}}\Big) } $$ for $n=0,1,\ldots,2000$, for the indicated index values, the following equalities hold $$ \sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} - \sqrt{\mathcal{T}_{n,8}}\Big) } = \left\{ \begin{array}{ll} (-1)^{k-1}\, \widehat{x}_{k}, &\quad \mbox{ for }\ n=3\, k-1 \geq 5,\\[0.8ex] \sqrt[3]{7}\, (-1)^{k}\, \widehat{y}_{k}, &\quad \mbox{ for }\ n=3\, k,\\[0.8ex] \sqrt[3]{49}\, (-1)^{k+1}\, \widehat{z}_{k}, &\quad \mbox{ for }\ n=3\, k+1,\\ \end{array} \right. $$ for $k=1,2,\ldots$, where \begin{align*} \widehat{x}_{1} &= 2, & \widehat{x}_{2} &= 5, & \widehat{x}_{3} &=16,\\ \widehat{y}_{1} &= 1, & \widehat{y}_{2} &= 4, & \widehat{y}_{3} &=12,\\ \widehat{z}_{1} &= 1, & \widehat{z}_{2} &= 3, & \widehat{z}_{3} &=9, \end{align*} and the elements of any of the sequences: $\{\widehat{x}_{k}\}_{k=1}^{\infty}$, $\{\widehat{y}_{k}\}_{k=1}^{\infty}$ and $\{\widehat{z}_{k}\}_{k=1}^{\infty}$, satisfy the following recurrent relation $$ \mathbb{X}_{n+3} - 4\, \mathbb{X}_{n+2} + 3\, \mathbb{X}_{n+1} +\mathbb{X}_{n} =0. $$ Also, the following interesting relationships occur (see~(\ref{aa})): \begin{align*} \widehat{x}_{k} &= A_{k}(-1) + 2\, C_{k}(-1) - C_{k-2}(-1),\qquad k\geq 2,\\ \widehat{y}_{k} &= A_{k}(-1) + C_{k}(-1),\\ \widehat{z}_{k} &= C_{k}(-1) - B_{k}(-1). \end{align*} Hence, by~\cite[eqs.~(3.17), (3.18), (3.19)]{WitulaSlota-A7}, we obtain, inter alia, the following Binet formulas: \begin{multline*} \widehat{x}_{k} = \big( 2 - 2\, \cos \alpha + 4\, \cos 2\alpha \big)\, \big( 1 - 2\, \cos \alpha \big)^{k-2} +{}\\ {}+ \big( 2 - 2\, \cos 2\alpha + 4\, \cos 4\alpha \big)\, \big( 1 - 2\, \cos 2\alpha \big)^{k-2} +{}\\ {}+ \big( 2 - 2\, \cos 4\alpha + 4\, \cos \alpha \big)\, \big( 1 - 2\, \cos 4\alpha \big)^{k-2}, \end{multline*} \begin{multline*} \widehat{y}_{k} = \frac{2}{7}\, \big( 1 + \cos 2\alpha - 2\, \cos 4\alpha \big)\, \big( 1 - 2\, \cos \alpha \big)^{k} +{}\\ {}+ \frac{2}{7}\, \big( 1 + \cos 4\alpha - 2\, \cos \alpha \big)\, \big( 1 - 2\, \cos 2\alpha \big)^{k} +{}\\ {}+ \frac{2}{7}\, \big( 1 + \cos \alpha - 2\, \cos 2\alpha \big)\, \big( 1 - 2\, \cos 4\alpha \big)^{k} , \end{multline*} \begin{multline*} \widehat{z}_{k} = \frac{2}{7}\, \big( \cos 2\alpha - \cos \alpha \big)\, \big( 1 - 2\, \cos \alpha \big)^{k} + {}\\ {}+ \frac{2}{7}\, \big( \cos 4\alpha - \cos 2\alpha \big)\, \big( 1 - 2\, \cos 2\alpha \big)^{k} +{}\\ {}+ \frac{2}{7}\, \big( \cos \alpha - \cos 4\alpha \big)\, \big( 1 - 2\, \cos 4\alpha \big)^{k}. \end{multline*} Next, as results from direct observation of the value of expression $$ \sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} + \sqrt{\mathcal{T}_{n,8}}\Big) } $$ for $n=0,1,\ldots,2000$, for the indicated index values, the following equations hold $$ \sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} + \sqrt{\mathcal{T}_{n,8}}\Big) } = \left\{ \begin{array}{ll} \widetilde{x}_{k}, &\quad \mbox{ for }\ n=3\, k+2,\\[0.8ex] \sqrt[3]{7}\, \widetilde{y}_{k}, &\quad \mbox{ for }\ n=3\, k+1,\\[0.8ex] \sqrt[3]{49}\, \widetilde{z}_{k}, &\quad \mbox{ for }\ n=3\, k,\\ \end{array} \right. $$ for $k=1,2,\ldots$, where $x_0=2$ and \begin{align*} \widetilde{x}_{1} &= 8, & \widetilde{x}_{2} &= 29, & \widetilde{x}_{3} &=120,\\ \widetilde{y}_{1} &= 1, & \widetilde{y}_{2} &= 2, & \widetilde{y}_{3} &=10,\\ \widetilde{z}_{1} &= 1, & \widetilde{z}_{2} &= 3, & \widetilde{z}_{3} &=13, \end{align*} and the elements of any of the sequences $\{\widetilde{x}_{k}\}_{k=2}^{\infty}$, $\{\widetilde{y}_{k}\}_{k=1}^{\infty}$ and $\{\widetilde{z}_{k}\}_{k=1}^{\infty}$ satisfy the following recurrent relation: $$ \mathbb{X}_{n+3} - 3\, \mathbb{X}_{n+2} - 4\, \mathbb{X}_{n+1} -\mathbb{X}_{n} =0. $$ After substitution $x\mapsto (x-1)$ in the respective characteristic polynomial of this relation, we obtain polynomial $x^3-7\, x-7$, for which by~\cite[eq.~(4.14)]{WitulaSlota-A7s}, for $n=1$, we obtain \begin{equation*} \mathbb{X}^3 -7 \, \mathbb{X} - 7 = \prod_{k=0}^{2} \Big( \mathbb{X} + \frac{\sqrt{7}}{2}\, \csc (2^k\alpha) \Big). \end{equation*} Hence, the following Binet formulas hold \begin{equation*} {p}_{n} = a_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc \alpha\big)^{\! n} + b_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc 2\alpha\big)^{\! n} + c_p\, \big( 1 - \frac{\sqrt{7}}{2}\, \csc 4\alpha\big)^{\! n}, \end{equation*} for every $p\in \{\widetilde{x},\widetilde{y},\widetilde{z}\}$, and where \begin{align*} a_{\tilde{x}} &\approx -1.246979604, & b_{\tilde{x}} &\approx 0.4450418679, & c_{\tilde{x}} &\approx 1.801937736,\\ a_{\tilde{y}} &\approx -1.064961507, & b_{\tilde{y}} &\approx 0.9189943261, & c_{\tilde{y}} &\approx 0.1459671806,\\ a_{\tilde{z}} &\approx -0.4355596199, & b_{\tilde{z}} &\approx 0.2417173531, & c_{\tilde{z}} &\approx 0.1938422668. \end{align*} Additionally, we note that \begin{align*} \widetilde{x}_{0} &= a_{\tilde{x}} + b_{\tilde{x}} + c_{\tilde{x}} = 1,\\ \widetilde{y}_{0} &= a_{\tilde{y}} + b_{\tilde{y}} + c_{\tilde{y}} = 0,\\ \widetilde{z}_{0} &= a_{\tilde{z}} + b_{\tilde{z}} + c_{\tilde{z}} = 0. \end{align*} By juxtaposing the obtained values of expressions $$ \sqrt[3]{ \frac{1}{2}\, \Big(\mathcal{S}_{n,8} \pm \sqrt{\mathcal{T}_{n,8}}\Big) } $$ we can now invest formula~(\ref{w6-d}) to the new interesting form (for cases $n=3k, 3k+1, 3k+2$ respectively, and only within the indicated range of values $n=1,2,\ldots,2000$): \begin{multline*} \sqrt[3]{2\, \cos \alpha}\, \big( 2\, \cos 2\alpha\big)^{k} + \sqrt[3]{2\, \cos 2\alpha}\, \big( 2\, \cos 4\alpha\big)^{k} + \sqrt[3]{2\, \cos 4\alpha}\, \big( 2\, \cos \alpha\big)^{k} ={}\\ {}= \sqrt[3]{ f_{3k} + 6 + 3\, (-1)^{k-1} \widehat{y}_{k}\, \sqrt[3]{7} - 3\, \widetilde{z}_{k}\, \sqrt[3]{49}}\,, \end{multline*} \begin{multline*} \sqrt[3]{\sec 4\alpha}\, \big( 2\, \cos 2\alpha \big)^{k} + \sqrt[3]{\sec \alpha}\, \big( 2\, \cos 4\alpha \big)^{k} + \sqrt[3]{\sec 2\alpha}\, \big( 2\, \cos \alpha \big)^{k} ={}\\ {}= \sqrt[3]{ 2\, \big(f_{3k+1} + 6 - 3\, \widetilde{y}_{k}\, \sqrt[3]{7} + 3\,(-1)^{k}\, \widehat{z}_{k}\, \sqrt[3]{49}\big)}\,, \end{multline*} and the formula which is equivalent to relation~(\ref{w-nr2}) and which generates the identity \begin{equation} 7\, \psi_{k}^{3} = f_{3k+2} + 6 - 3\, \big( \widetilde{x}_{k} + (-1)^{k}\, \widehat{x}_{k+1} \big). \end{equation} \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{9.} By~\cite[eq.~(4.15)]{WitulaSlota-A7s} we obtain \begin{multline}\label{w7-d} \big( 2\, \sin \alpha \big)^{n/3} + \big( 2\, \sin 2\alpha \big)^{n/3} + \big( 2\, \sin 4\alpha \big)^{n/3} = {}\\ {}= \sqrt[3]{ z_{n-1} + 6\, (-1)^{n}\, 7^{n/6} - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n,9} + \sqrt{\mathcal{T}_{n,9}} } + \sqrt[3]{ \mathcal{S}_{n,9} - \sqrt{\mathcal{T}_{n,9}} }\,\, \Big) }, \end{multline} where \begin{align} \mathcal{S}_{n,9} &= \big( \tfrac{1}{2}\, z_{n-1} + 3\, (-1)^{n}\, 7^{n/6} \big)\, \big( z_{2n-1} - z_{n-1}^{2} \big) - \nonumber\\ &\phantom{==} - 6\cdot 7^{n/3} \, z_{n-1} - 9\, (-1)^{n}\, 7^{n/2},\label{w7-e}\\ \mathcal{T}_{n,9} &= (-1)^{n}\, 7^{n/2}\, z_{n-1}\, \big( 5\, z_{n-1}^{2} - 9\, z_{2n-1} \big) +\nonumber\\ &\phantom{==} + \tfrac{1}{4}\, \big( 2\, z_{2n-1} - z_{n-1}^{2} \big)\, \big( z_{n-1}^{2} - z_{2n-1} \big)^{2} - 27 \cdot 7^{n}, \label{w7-f}\\ %\end{align} %$$ p_{n,9}&=-z_{n-1},\quad q_{n,9}= \tfrac{1}{2} \, \big( {z}_{n-1}^{2} - z_{2n-1} \big),\quad r_{n,9}= (-1)^{n-1}\, 7^{n/2}. %$$ \end{align} %\begin{align} %p_{n,7}&=-z_{n-1},\label{w7-a}\\ %q_{n,7}&= \tfrac{1}{2} \, \big( {z}_{n-1}^{2} - z_{2n-1} \big),\label{w7-b}\\ %r_{n,7} &= (-1)^{n-1}\, 7^{n/2}.\label{w7-c} %\end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{10.} By~\cite[eq.~(2.2)]{WitulaSlota-A7s} we have (for $\delta\in \mathbb{R}$): \begin{multline}\label{ram-2-w6} \big( 1 + 2\, \delta\, \cos \alpha \big)^{n/3} + \big( 1 + 2\, \delta\, \cos 2\alpha \big)^{n/3} + \big( 1 + 2\, \delta\, \cos 4\alpha \big)^{n/3} = {}\\ {}= \bigg[ \mathcal{A}_{n}(\delta) + 6\, \widehat{\delta}^{\,n/3} - \frac{3}{\sqrt[3]{2}} \, \Big( \sqrt[3]{ \mathcal{S}_{n}(\delta) + \sqrt{\mathcal{T}_{n}(\delta)} } + \sqrt[3]{ \mathcal{S}_{n}(\delta) - \sqrt{\mathcal{T}_{n}(\delta)} }\,\, \Big) \bigg]^{1/3}, \end{multline} where \begin{align} \widehat{\delta} &= \delta^3-2\, \delta^2-\delta+1,\label{ram-2-w7}\\ \mathcal{S}_{n}(\delta) &= -\mathcal{A}_{n}(\delta)\, \mathcal{B}_{n}(\delta) - 6\, \mathcal{B}_{n}(\delta)\, \widehat{\delta}^{\,n/3} - 6\, \mathcal{A}_{n}(\delta)\, \widehat{\delta}^{\,2n/3} - 9\, \widehat{\delta}^{\,n},\label{ram-2-w8}\\ \mathcal{T}_{n}(\delta) &= \mathcal{A}_{n}^{2}(\delta)\,\mathcal{B}_{n}^{2}(\delta) - 4\, \mathcal{A}_{n}^{3}(\delta)\, \widehat{\delta}^{\,n} - 4\,\mathcal{B}_{n}^{3}(\delta)%\nonumber\\ %&\phantom{==} + 18\, \mathcal{A}_{n}(\delta)\,\mathcal{B}_{n}(\delta)\, \widehat{\delta}^{\,n} - 27\, \widehat{\delta}^{\,2\, n}\label{ram-2-w9aa}\\ &= \big(\mathcal{A}_{n}(\delta)\,\mathcal{B}_{n}(\delta) + 9\, \widehat{\delta}^{\,n} \big)^2 -4\, \big( \mathcal{A}_{n}^{3}(\delta)\, \widehat{\delta}^{\,n} +\mathcal{B}_{n}^{3}(\delta) \big) - 108\, \widehat{\delta}^{\,2\, n}.\label{ram-2-w9} \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{11.} By~\cite[eq.~(6.14)]{WitulaSlota-A7s} we have \begin{multline}\label{ram-2-w10} 7^{n/6}\, \big( ( \cot \alpha )^{n/3}\,+ ( \cot 2\alpha )^{n/3}\,+ ( \cot 4\alpha )^{n/3} \big) = {}\\ {}= \bigg[ 3^n\, \mathcal{A}_{n}(\tfrac{2}{3}) + 6\, (-1)^{n}\, 7^{n/3} + \frac{3}{\sqrt[3]{2}} \, 7^{n/3}\, \Big( \sqrt[3]{ \mathcal{S}_{n}^{'} + \sqrt{\mathcal{T}_{n}^{'}} } + \sqrt[3]{ \mathcal{S}_{n}^{'} - \sqrt{\mathcal{T}_{n}^{'}} }\,\, \Big) \bigg]^{1/3}, \end{multline} where \begin{align} \mathcal{S}_{n}^{'} &= \big( 3^n\, \mathcal{A}_{n}(\tfrac{2}{3}) + 6\,(-1)^{n}\, 7^{n/3} \big)\, \Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big) %+ {}\nonumber\\ %&\phantom{==} + 6\, \big( \tfrac{3}{\sqrt[3]{7}} \big)^{n}\, \mathcal{A}_{n}(\tfrac{2}{3}) + 9\, (-1)^{n}, \label{ram-2-w11}\\ \mathcal{T}_{n}^{'} &= %3^{2n}\, \mathcal{A}_{n}^{2}(\tfrac{2}{3})\, \Omega_{n}^{2} \big( \tfrac{2\, i}{\sqrt{7}}\big) %+18\, (-3)^{n}\, \mathcal{A}_{n}(\tfrac{2}{3})\, \Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big) -{}\nonumber\\ %&\phantom{==} %- 4\, \big( 7^n\, \Omega_{n}^{3} \big( \tfrac{2\, i}{\sqrt{7}}\big) %+ (-27)^{n}\, 7^{-n}\, \mathcal{A}_{n}^{3}(\tfrac{2}{3}) \big) -27 \nonumber\\ %&= \big( (-3)^{n}\, \mathcal{A}_{n}(\tfrac{2}{3})\, \Omega_{n} \big( \tfrac{2\, i}{\sqrt{7}}\big) +9 \big)^{2} %- {}\nonumber\\ %&\phantom{==} - 4\, \big( 7^n\, \Omega_{n}^{3} \big( \tfrac{2\, i}{\sqrt{7}}\big) + \big(-\tfrac{27}{7} \big)^{n}\, \mathcal{A}_{n}^{3}(\tfrac{2}{3}) \big) - 108. \label{ram-2-w12} \end{align} The numbers $\Omega_{n}(\delta)$ are defined for $n\in \mathbb{N}$ and $\delta\in \mathbb{C}$, in the following way $$ \Omega_{n}(\delta) := \big( 1+2\, i\, \delta\, \sin\alpha \big)^{n} + \big( 1+2\, i\, \delta\, \sin 2\alpha \big)^{n} + \big( 1+2\, i\, \delta\, \sin 4\alpha \big)^{n}, $$ (see~\cite[Section~6]{WitulaSlota-A7s} for more details). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{remark} Moreover, we have \begin{multline}\label{gwiazdka} \big(\mathbb{X}- ( \tan \alpha )^{n}\big) \big(\mathbb{X}- ( \tan 2\alpha )^{n}\big) \big(\mathbb{X}- ( \tan 4\alpha )^{n}\big) = {}\\ {} = \mathbb{X}^3 - ({-}\sqrt{7} )^{n} \, \Omega_{n} \big(\tfrac{2\, i}{\sqrt{7}}\big)\, \mathbb{X}^2 + (-3)^n\, \mathcal{A}_{n}(\tfrac{2}{3})\, \mathbb{X} - ({-}\sqrt{7})^{n}. \end{multline} This "distribution" easily results from~\cite[eq.~(6.14)]{WitulaSlota-A7s}. Now we will present a~direct proof of the relation~(\ref{gwiazdka}), because the formula~(6.14) in~\cite{WitulaSlota-A7s} is presented without a~proof. For this purpose, let us suppose that $\xi = \exp ( i\, 2\, \pi/7)$. Then we have \begin{multline*} ( \tan \alpha )^{n}+ ( \tan 2\alpha )^{n}+ ( \tan 4\alpha )^{n} ={}\\ {}= \Big( {-}i\, \frac{\xi - \xi^6}{\xi + \xi^6} \Big)^n + \Big( {-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5} \Big)^n + \Big( {-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3} \Big)^n ={}\\ {}= \Big( \frac{-i}{(\xi + \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3)} \Big)^{n}\, \bigg[ \Big( (\xi - \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3) \Big)^{n} +{}\\ {}+ \Big( (\xi^2 - \xi^5)(\xi + \xi^6)(\xi^4 + \xi^3) \Big)^{n} + \Big( (\xi^4 - \xi^3)(\xi + \xi^6)(\xi^2 + \xi^5) \Big)^{n} \bigg]={}%\\ \end{multline*} \begin{multline*} {}\stackrel{\mbox{\small \cite[eq.~(1.4)]{WitulaSlota-A7s}}}{=} (-i)^n\, \bigg[ \Big( 2\,(\xi^2 - \xi^5) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n} +{}\\ {}+ \Big( 2\,(\xi^4 - \xi^3) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n} +{}\\ {}+ \Big( 2\,(\xi - \xi^6) - (\xi - \xi^6) -(\xi^2 - \xi^5) - (\xi^4 - \xi^3) \Big)^{n} \bigg] ={}\\ {} \stackrel{\mbox{\small \cite[eq.~(1.1)]{WitulaSlota-A7s}}}{=} \bigg[ \Big( {-}2\, i\, (\xi^2-\xi^5) - \sqrt{7} \Big)^{n} + \Big( {-}2\, i\, (\xi^4-\xi^3) - \sqrt{7} \Big)^{n} +{}\\ {}+ \Big( {-}2\, i\, (\xi-\xi^6) - \sqrt{7} \Big)^{n} \bigg] = \big( {-}\sqrt{7} \big)^{n}\, \Omega_{n} \big(\tfrac{2\, i}{\sqrt{7}}\big), \end{multline*} and \begin{multline*} \big( \tan \alpha\,\tan 2\alpha \big)^{n}+ \big( \tan \alpha\,\tan 4\alpha \big)^{n}+ \big( \tan 2\alpha\,\tan 4\alpha \big)^{n} ={}\\ {}= \Big( \Big({-}i\, \frac{\xi - \xi^6}{\xi + \xi^6}\Big)\, \Big({-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5}\Big) \Big)^n + \Big( \Big({-}i\, \frac{\xi - \xi^6}{\xi + \xi^6}\Big)\, \Big({-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3}\Big) \Big)^n +{}\\ {}+ \Big( \Big({-}i\, \frac{\xi^2 - \xi^5}{\xi^2 + \xi^5}\Big)\, \Big({-}i\, \frac{\xi^4 - \xi^3}{\xi^4 + \xi^3}\Big) \Big)^n ={}\\ %\end{multline*} %\begin{multline*} {}= \Big((\xi + \xi^6)(\xi^2 + \xi^5)(\xi^4 + \xi^3)\Big)^{-n}\, \bigg[ \Big( (\xi^6 - \xi)(\xi^2 - \xi^5)(\xi^4 + \xi^3) \Big)^{n} +{}\\ {}+ \Big( (\xi^6 - \xi)(\xi^4 - \xi^3)(\xi^2 + \xi^5) \Big)^{n} + \Big( (\xi^5 - \xi^2)(\xi^4 - \xi^3)(\xi + \xi^6) \Big)^{n} \bigg] ={}\\ {}= \Big(\! {-}3 - 2\, (\xi+\xi^6)\! \Big)^{n} + \Big(\! {-}3 - 2\, (\xi^4+\xi^3)\! \Big)^{n} + \Big(\! {-}3 - 2\, (\xi^2+\xi^5)\! \Big)^{n} = (-3)^n\, \mathcal{A}_{n}(\tfrac{2}{3}). \end{multline*} \end{remark} \noindent \textbf{Final remark.} I~was only after I~received the referee report on my paper that I~learnt about two important publications in this field~\cite{Shevelev2007,Shevelev2009}. Certainly, both papers supplement and enrich the contents of Section~\ref{roz3a}. As a~spontaneous reaction to~\cite{Shevelev2007} and the report on the present paper, two more papers sprang up~\cite{Witula-RCP} and~\cite{Witula-Sup}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Acknowledgments} The author wish to express their gratitude to the Reviewer for several helpful comments concerning the first version of my paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{table}[htb!] \begin{center} {\small% \caption{}\label{tabela} \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 $\psi_{n}$ & $-1$ & $0$ & $-3$ & $2$ & $-8$ & $9$ & $-23$ & $33$ & $-70$ & $113$ & $-220$ & $376$\\ \hline $\varphi_{n}$ & $0$ & $-1$ & $1$ & $-3$ & $4$ & $-9$ & $14$ & $-28$ & $47$ & $-89$ & $155$ & $-286$\\\hline\hline \end{tabular} } \end{center} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{10} \bibitem{Berndt} B.~C.~Berndt, \textit{Ramanujan's Notebooks, Part~IV}, Springer, 1994. %New York \bibitem{Berndt2} B.~C.~Berndt, H.~H.~Chan and L.-Ch.~Zhang, Radicals and units in Ramanujan's work, \textit{Acta Arith.} \textbf{87} (1998), 145--158. \bibitem{Berndt3} B.~C.~Berndt, Y.-S.~Choi and S.-Y.~Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, in B.~C.~Berndt and F.~Gesztesy, eds., \textit{Continued Fractions: From Analytic Number Theory to Constructive Approximation}, AMS, 1999, pp.~15--56. %Providence \bibitem{GrzymkowskiWitula} R.~Grzymkowski and R.~Witu{\l}a, \textit{Calculus Methods in Algebra, Part One}, WPKJS, 2000 (in Polish). \bibitem{Kreczmar} W.~A.~Kreczmar, \textit{A~Collection of Problems in Algebra}, Nauka, 1961. %Moscow \bibitem{Modenov} P.~S.~Modenov, \textit{Collection of Problems for a Special Course of Elementary Mathematics}, Sovetskaja Nauka, 1957. %Moscow \bibitem{Shevelev1988} V.~S.~Shevelev, {Three Ramanujan's formulas}, \textit{Kvant} \textbf{6} (1988), 52--55 (in Russian). \bibitem{Shevelev2007} V.~Shevelev, {On Ramanujan cubic polynomials}, \htmladdnormallink{\textit{arXiv:0711.3420v1}}{http://arxiv.org/abs/0711.3420}, 2007. \bibitem{Shevelev2009} V.~Shevelev, {Gold ratio and a~trigonometric identity}, \htmladdnormallink{\textit{arXiv:0909.5072v1}}{http://arxiv.org/abs/0909.5072}, 2009. \bibitem{Sloane} N.~J.~A.~Sloane, \htmladdnormallink{\textit{The On-Line Encyclopedia of Integer Sequences}} {http://www.research.att.com/$\sim$njas/sequences/}, 2009. %{The On-Line Encyclopedia of Integer Sequences},\\ %{http://www.research.att.com/$\sim$njas/sequences/} (2009). \bibitem{WitulaSlota-A7} R.~Witu{\l}a, D.~S{\l}ota and A.~Warzy{\'n}ski, {Quasi-Fibonacci numbers of the seventh order}, \textit{J.~Integer Seq.} \textbf{9} (2006), \htmladdnormallink{Article 06.4.3}{http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html}. \bibitem{WitulaSlota-A7s} R.~Witu{\l}a and D.~S{\l}ota, {New Ramanujan-type formulas and quasi-Fibonacci numbers of order~7}, \textit{J.~Integer Seq.} \textbf{10} (2007), \htmladdnormallink{Article 07.5.6}{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html}. \bibitem{WitulaSlota-A11} R.~Witu{\l}a and D.~S{\l}ota, {Quasi-Fibonacci numbers of order~11}, \textit{J.~Integer Seq.} \textbf{10} (2007), \htmladdnormallink{Article 07.8.5}{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.html}. \bibitem{Witula-RCP} R.~Witu{\l}a, {Full description of Ramanujan cubic polynomials}, submitted. \bibitem{Witula-Sup} R.~Witu{\l}a, {Ramanujan type trigonometric formulas for arguments $\frac{2\, \pi}{7}$ and $\frac{2\, \pi}{9}$~-- supplement}, submitted. \bibitem{Yaglom} A.~M.~Yaglom and I.~M.~Yaglom, {An elementary derivation of the formulas of Wallis, Leibnitz and Euler for the number~$\pi$}, \textit{Uspehi Matem. Nauk} \textbf{8} (1953), 181--187. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11B37; Secondary 11B83, 11Y55, 33B10. \noindent \emph{Keywords:} Ramanujan equalities, trigonometric recurrences. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A094648} and \seqnum{A006053}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received July 6 2009; Revised version received December 3 2009. Published in {\it Journal of Integer Sequences}, December 3 2009. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in \end{document}