A157877 Expansion of (1-x)*x/(x^2-30*x+1).

1, 29, 869, 26041, 780361, 23384789, 700763309, 20999514481, 629284671121, 18857540619149, 565096933903349, 16934050476481321, 507456417360536281, 15206758470339607109, 455695297692827676989, 13655652172314490702561, 409213869871741893399841

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.

A157877 is the a(n) sequence for A=7.

Positive values of x (or y) satisfying x^2 - 30xy + y^2 + 28 = 0. - Colin Barker, Feb 23 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (30,-1).

Crossrefs

Cf. A157879, A157877, A157878.

Cf. similar sequences listed in A238379.

Programs

Magma

I:=[1,29,869]; [n le 3 select I[n] else 30*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 25 2014

Mathematica

LinearRecurrence[{30,-1},{1,29},30] (* Harvey P. Dale, Dec 14 2011 *)
CoefficientList[Series[(1 - x)/(x^2 - 30 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 25 2014 *)

PARI

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

Formula

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

a(1)=1, a(2)=29; for n>2, a(n) = 30*a(n-1)-a(n-2).

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

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

a(n) = (1/8)*(4-sqrt(14))*(1+(15+4*sqrt(14))^(2*n-1))/(15+4*sqrt(14))^(n-1). - Bruno Berselli, Feb 25 2014

From Andrea Pinos, Oct 05 2022: (Start)

a(n) = ceiling((C^n)/(C+1)), where C = 15 + 4*sqrt(14) = sqrt(225) + sqrt(224).

Limit_{n->oo} a(n+1)/a(n) = C. (End)

Extensions

Edited by Alois P. Heinz, Sep 09 2011