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# User:L. Edson Jeffery/LA212329

A212329 (LaTeX version): Expansion of ${\displaystyle {\frac {x(5+x)}{1-7x+7x^{2}-x^{3}}}}$.

## Sequence data

0, 5, 36, 217, 1272, 7421, 43260, 252145, 1469616, 8565557, 49923732, 290976841, 1695937320, 9884647085, 57611945196, 335787024097, 1957110199392, 11406874172261, 66484134834180, 387497934832825

## Offset

1, 2

 Table of differences re Table A182441.
 This is a sequence of differences between rows ${\displaystyle k}$ and ${\displaystyle k+1}$ of table A182441. That is if A182441${\displaystyle (k+1,0)}$ ${\displaystyle -}$ A182441${\displaystyle (k,0)=1}$, ${\displaystyle a(n)}$ ${\displaystyle =}$ A182441${\displaystyle (k+1,n+1)}$ ${\displaystyle -}$ A182441${\displaystyle (k,n+1)}$ for ${\displaystyle n=0}$ to ${\displaystyle 3}$. The remainder of the sequence is a continuation using the recursive formula ${\displaystyle D(n)=6D(n-1)-D(n-2)+6}$.
 It appears that for ${\displaystyle n>0}$, ${\displaystyle a(n)}$ is divisible by A213005${\displaystyle (n)}$.
 It appears that if ${\displaystyle p}$ is a prime of the form ${\displaystyle 8r\pm 1}$, ${\displaystyle a(p-1)\equiv 0{\pmod {p}}}$; and that if ${\displaystyle p}$ is a prime of the form ${\displaystyle 8r\pm 3}$, ${\displaystyle a(p+1)\equiv 0{\pmod {p}}}$.

## Formulas

${\displaystyle a(n)=7a(n-1)-7a(n-2)+a(n-3)}$.

nonn

## Author(s)

Kenneth J Ramsey, May 14 2012

## Missing %-fields

For links, references, programs, cross-references, extensions or other missing %-fields (if they exist), see A212329.

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