User:L. Edson Jeffery/LA212329

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

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

Comments

Table of differences re Table A182441.

This is a sequence of differences between rows $k$ and $k+1$ of table A182441. That is if A182441 $(k+1,0)$ $-$ A182441 $(k,0)=1$ , $a(n)$ $=$ A182441 $(k+1,n+1)$ $-$ A182441 $(k,n+1)$ for $n=0$ to $3$ . The remainder of the sequence is a continuation using the recursive formula $D(n)=6D(n-1)-D(n-2)+6$ .

It appears that for $n>0$ , $a(n)$ is divisible by A213005 $(n)$ .

It appears that if $p$ is a prime of the form $8r\pm 1$ , $a(p-1)\equiv 0{\pmod {p}}$ ; and that if $p$ is a prime of the form $8r\pm 3$ , $a(p+1)\equiv 0{\pmod {p}}$ .