A212329 Expansion of x*(5+x)/(1-7*x+7*x^2-x^3).

0, 5, 36, 217, 1272, 7421, 43260, 252145, 1469616, 8565557, 49923732, 290976841, 1695937320, 9884647085, 57611945196, 335787024097, 1957110199392, 11406874172261, 66484134834180, 387497934832825, 2258503474162776, 13163522910143837, 76722633986700252

Offset: 1

Author

Kenneth J Ramsey, May 14 2012

Keywords

• nonn
• easy

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 8*r +/- 1 then a(p-1) == 0 (mod p), and that if p is a prime of the form 8*r +/- 3 then a(p+1) == 0 (mod p).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A182441, A213005.

Programs

Mathematica

m = 12; n = 1; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m + 6; Sow[t];m = n; n = t;c++]][[2,1]]
CoefficientList[ Series[x (5 + x)/(1 - 7x + 7x^2 - x^3), {x, 0, 20}], x] (* or *)
LinearRecurrence[{7, -7, 1}, {0, 5, 36}, 21] (* Robert G. Wilson v, Jun 24 2014 *)

PARI

concat(0, Vec(x^2*(5+x)/((1-x)*(1-6*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 05 2016

Formula

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).

From Colin Barker, Mar 05 2016: (Start)

a(n) = (-6+(5-3*sqrt(2))*(3+2*sqrt(2))^n + (3-2*sqrt(2))^n*(5+3*sqrt(2)))/4.

G.f.: x*(5+x) / ((1-x)*(1-6*x+x^2)).

(End)