A157874 Expansion of 104*x^2 / (-x^3+675*x^2-675*x+1).

0, 104, 70200, 47314800, 31890105104, 21493883525400, 14486845606014600, 9764112444570315104, 6580997300794786365600, 4435582416623241440099400, 2989575967806763935840630104, 2014969766719342269515144590800, 1358086633192868882889271613569200

Offset: 1

Author

Paul Weisenhorn, Mar 08 2009

Keywords

• nonn
• easy

Comments

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:

A * c(n)+1 = a(n)^2,

(A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.

A157874 is the c(n) sequence for A=6.

Links

• Colin Barker, Table of n, a(n) for n = 1..350
• Index entries for linear recurrences with constant coefficients, signature (675,-675,1).

Crossrefs

6*A157874(n)+1 = A153111(n)^2.

7*A157874(n)+1 = A157461(n)^2.

Programs

Mathematica

Rest[CoefficientList[Series[104x^2/(-x^3+675x^2-675x+1),{x,0,20}],x]] (* or *) LinearRecurrence[{675,-675,1},{0,104,70200},20] (* Harvey P. Dale, Oct 04 2015 *)

PARI

concat(0, Vec(104*x^2/(-x^3+675*x^2-675*x+1) + O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012

PARI

a(n) = -round((337+52*sqrt(42))^(-n)*(-1+(337+52*sqrt(42))^n)*(13+2*sqrt(42)+(-13+2*sqrt(42))*(337+52*sqrt(42))^n))/168 \\ Colin Barker, Jul 25 2016

Formula

G.f.: 104*x^2 / (-x^3+675*x^2-675*x+1).

c(1) = 0, c(2) = 104, c(3) = 675*c(2), c(n) = 675 * (c(n-1)-c(n-2)) + c(n-3) for n>3.

a(n) = -((337+52*sqrt(42))^(-n)*(-1+(337+52*sqrt(42))^n)*(13+2*sqrt(42)+(-13+2*sqrt(42))*(337+52*sqrt(42))^n))/168. - Colin Barker, Jul 25 2016

Extensions

Edited by Alois P. Heinz, Sep 09 2011