\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\usepackage{rotating}


\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}

\section{Addendum}

These results - computed since publication - are additional to those reported in Appendix A above.

\bigskip

$\mathbf{h=2}$

\smallskip

\begin{tabular}{r@{\hspace{.15in}}r@{\hspace{.15in}}rrrrrrrrrrrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(2,k)$}&&$a_i$\\
23 & 196 && 1&3&4&6&10&13&15&21&29&37&45&53\\
 & && & & &61&69&77&85&89&91&94&95&98&100&104\\
23 & 196 && 1&3&4&6&10&13&15&21&29&37&45&53\\
 & && & & &61&69&77&85&89&91&94&95&96&98&100\\
23 & 196 && 1&3&4&6&10&13&15&21&29&37&45&53\\
 & && & & &61&69&77&83&85&88&92&94&95&97&98\\
\end{tabular}

\bigskip

$\mathbf{h=3}$

\smallskip

\begin{tabular}{r@{\hspace{.15in}}r@{\hspace{.15in}}rrrrrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(3,k)$}&&$a_i$\\
15 & 385 && 1&3&8&12&19&25&34&36&57&98&118&128&168&178&198\\
15 & 385 && 1&4&5&16&18&29&37&43&52&78&98&148&158&178&188\\
\end{tabular}

\bigskip

$\mathbf{h=4}$

\smallskip

\begin{tabular}{r@{\hspace{.15in}}r@{\hspace{.15in}}rrrrrrrrrrrrr}
$k$&\multicolumn{1}{c}{$n(4,k)$}&&$a_i$\\
12 & 700 &&  1&5&8&20&22&29&45&106&174&240&311&321\\
\end{tabular}

\bigskip
\pagebreak

$\mathbf{k=4}$

\smallskip


For $55 \leq h \leq 302$, $n(h,4)$ and $a_i$ are given by one of the following three sets of formulae:

\bigskip

$\begin{array}{rrcl}
        (A): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (4t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (2t + c_{51}) + ( t + c_{52})a_2 + (6t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

$\begin{array}{rrcl}
        (B): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (2t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (4t + c_{51}) + (3t + c_{52})a_2 + (2t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

$\begin{array}{rrcl}
        (C): & a_2 & = & (9t + c_{21})\\
        & a_3 & = & (4t + c_{31}) + (3t + c_{32})a_2\\
        & a_4 & = & (7t + c_{41}) + (2t + c_{42})a_2 + (2t + c_{43})a_3\\
        & n(h,4) & = & (t + c_{51}) + (4t + c_{52})a_2 + (6t + c_{53})a_3 + (3t + c_{54})a_4
\end{array}$

\bigskip

where  $h = 12t + r$,  $0 \leq r \leq 11$,  and $c_{ij}$ are given in the following table:


\vspace{-.1in}
\begin{center}
\begin{tabular}{rrrrrrrrrrrrrr}
$r$&&&$c_{21}$&$c_{31}$&$c_{32}$&$c_{41}$&$c_{42}$&$c_{43}$&$c_{51}$&$c_{52}$&$c_{53}$&$c_{54}$&Valid for:\\
0&A&&2&1&0&1&0&1&$-3$&0&4&$-1$&$ 4 \leq t \leq  5$\\
0&A&&1&0&0&0&0&0&$-2$&0&1&1&$ 6 \leq t \leq 11$\\
0&B&&2&2&$-1$&3&$-1$&0&$-1$&$-2$&$-1$&4&$12 \leq t \leq 25$\\
1&A&&1&0&2&1&1&0&0&0&1&0&$ 5 \leq t \leq 25$\\
2&A&&2&1&1&1&1&1&$-3$&1&4&0&$ 5 \leq t \leq  6$\\
2&A&&1&0&2&1&1&0&0&0&1&1&$ 7 \leq t \leq 20$\\
2&B&&5&3&$-1$&6&$-1$&0&0&$-2$&$-1$&5&$21\leq t\leq25$\\
3&A&&3&1&2&2&1&1&$-1$&0&4&0&$ 1 \leq t \leq 24$\\
4&A&&3&1&2&2&1&1&$-1$&0&4&1&$ 2 \leq t \leq 24$\\
5&A&&3&1&2&2&1&1&$-1$&0&4&2&$ 4 \leq t \leq 24$\\
6&A&&3&1&2&2&1&1&$-1$&0&4&3&$ 5 \leq t \leq 24$\\
7&A&&7&3&2&5&1&2&$-1$&0&7&1&$ 2 \leq t \leq 11$\\
7&A&&8&4&1&7&1&0&0&1&1&5&$12 \leq t \leq 24$\\
8&A&&7&3&3&5&2&2&$-1$&1&7&1&$1 \leq t \leq 16$\\
8&A&&8&4&1&7&1&0&0&1&1&6&$	17 \leq t \leq 24$\\
9&A&&7&3&3&5&2&2&$-1$&1&7&2&$	1 \leq t \leq 21$\\
9&A&&8&4&1&7&1&0&0&1&1&7&$22 \leq t \leq 24$\\
10&A&&7&3&3&5&2&2&$-1$&1&7&3&$	4 \leq t \leq 19$\\
10&C&&11&6&1&10&1&0&0&3&0&7&$ 20\leq t\leq24$\\   
11&A&&10&4&3&7&2&2&0&1&7&3&$	2 \leq t \leq  7$\\
11&B&&11&4&2&10&1&2&3&1&1&6&$ 8 \leq t \leq 22$\\
11&A&&11&5&2&9&2&0&1&2&1&7&$t = 23$\\
11&B&&12&5&1&12&1&0&3&0&-1&10&$  24 \leq t \leq 24$\\
\end{tabular}
\end{center}

\bigskip

$\mathbf{k=5}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrr}
$h$&\multicolumn{1}{c}{$n(h,5)$} && $a_1$&$a_2$&$a_3$&$a_4$&$a_5$\\
68 & 2330896 && 1&47&1000&16255&123331\\
69 & 2496702 && 1&53&752&16196&139500\\
70 & 2653201 && 1&52&789&15540&133254\\
71 & 2846834 && 1&53&804&16640&143028\\
72 & 3047485 && 1&57&866&17811&153105\\
73 & 3250580 && 1&55&910&17943&171807\\
74 & 3429203 && 1&58&960&18929&181248\\
75 & 3629795 && 1&59&1013&18977&182285\\
76 & 3864527 && 1&63&980&20223&194059\\
77 & 4103963 && 1&56&906&20468&196209\\
78 & 4416370 && 1&59&1013&21001&201513\\
79 & 4643287 && 1&62&1065&22080&211867\\
80 & 4975426 && 1&65&1141&23658&227046\\
81 & 5223883 && 1&63&1145&23756&227980\\
82 & 5519971 && 1&66&1200&24898&263837\\
83 & 5796515 && 1&62&1150&25011&264322\\
84 & 6139689 && 1&60&1090&26793&256691\\
85 & 6513282 && 1&60&1090&26853&284084\\
86 & 6912409 && 1&69&1255&28480&301525\\
87 & 7258582 && 1&63&1168&29884&316651\\
88 & 7677138 && 1&72&1336&31584&334950\\
89 & 8029729 && 1&67&1285&31617&366107\\
90 & 8525267 && 1&70&1412&32085&371775\\
\end{tabular}

\bigskip

$\mathbf{k=6}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrr}
$h$&\multicolumn{1}{c}{$n(h,6)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$&$a_6$\\
26 & 156744 && 1&19&177&816&6708&18060\\
\end{tabular}

\bigskip

$\mathbf{k=7}$

\smallskip

\begin{tabular}{r@{\hspace{.2in}}r@{\hspace{.2in}}rrrrrrrr}
$h$&\multicolumn{1}{c}{$n(h,7)$}&&$a_1$&$a_2$&$a_3$&$a_4$&$a_5$&$a_6$&$a_7$\\
14 & 24466 && 1&12&52&225&546&3033&5464\\
\end{tabular}

\pagebreak


\end{document}