A157878 Expansion of x*(1+x)/(x^2-30*x+1).

1, 31, 929, 27839, 834241, 24999391, 749147489, 22449425279, 672733610881, 20159558901151, 604114033423649, 18103261443808319, 542493729280825921, 16256708616980969311, 487158764780148253409, 14598506234787466632959, 437468028278843850735361

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 2 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.

this is the b(n) sequence for A=7.

Links

• Robert Israel, Table of n, a(n) for n = 1..610
• Index entries for linear recurrences with constant coefficients, signature (30,-1).

Crossrefs

7*A157879(n)+1 = A157877(n)^2.

8*A157879(n)+1 = A157878(n)^2.

Programs

Maple

f:= gfun:-rectoproc({a(1) = 1, a(2) = 31, a(n) = 30*a(n-1)-a(n-2)}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Jul 09 2015

Mathematica

CoefficientList[Series[x*(1 + x)/(x^2 - 30 x + 1), {x, 0, 17}], x] (* Michael De Vlieger, Jul 09 2015 *)
LinearRecurrence[{30,-1},{1,31},20] (* Harvey P. Dale, Sep 05 2021 *)

PARI

Vec((1+x)/(x^2-30*x+1)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

G.f.: x*(1+x)/(x^2-30*x+1).

a(1) = 1, a(2) = 31, a(n) = 30*a(n-1)-a(n-2) for n>2.

a(n) = ((15-4*sqrt(14))^(n-1)*(7-2*sqrt(14))+(7+2*sqrt(14))*(15+4*sqrt(14))^(n-1))/14. - Gerry Martens, Jul 09 2015

Extensions

Edited by Alois P. Heinz, Sep 09 2011