A157459 Expansion of 72*x^2 / (1 - 323*x + 323*x^2 - x^3).

0, 72, 23256, 7488432, 2411251920, 776415629880, 250003421569512, 80500325329753056, 25920854752758914592, 8346434730063040745640, 2687526062225546361181560, 865375045601895865259716752, 278648077157748243067267612656, 89723815469749332371794911558552

Offset

1,2

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.

A157459 is the c(n) sequence for A=4.

Links

Colin Barker, Table of n, a(n) for n = 1..250

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Formula

4*a(n) + 1 = A007805(n-1)^2.

5*a(n) + 1 = A049629(n-1)^2.

G.f.: 72*x^2/(1 - 323*x + 323*x^2 - x^3).

c(1) = 0, c(2) = 72, c(3) = 323*c(2), c(n) = 323*(c(n-1) - c(n-2)) + c(n-3) for n>3.

a(n) = -((161+72*sqrt(5))^(-n)*(-1+(161+72*sqrt(5))^n)*(9+4*sqrt(5)+(-9+4*sqrt(5))*(161+72*sqrt(5))^n))/80. - Colin Barker, Jul 25 2016

a(n) = 72*A298271(n-1). - Greg Dresden, Dec 02 2021

a(n) = 2*A201003(n-1). - Amiram Eldar, Dec 01 2024

Mathematica

LinearRecurrence[{323, -323, 1}, {0, 72, 23256}, 20] (* Harvey P. Dale, Feb 28 2021 *)

Prog

(PARI) concat(0, Vec(72*x^2/(1-323*x+323*x^2-x^3)+O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012
(PARI) a(n) = -round((161+72*sqrt(5))^(-n)*(-1+(161+72*sqrt(5))^n)*(9+4*sqrt(5)+(-9+4*sqrt(5))*(161+72*sqrt(5))^n))/80 \\ Colin Barker, Jul 25 2016

Crossrefs

Cf. A007805, A049629, A157014, A157459, A201003, A298271.

Sequence in context: A373650 A243234 A246633 * A159393 A151627 A223124

Adjacent sequences: A157456 A157457 A157458 * A157460 A157461 A157462

Keyword

nonn,easy

Author

Paul Weisenhorn, Mar 01 2009

Extensions

Edited by Alois P. Heinz, Sep 09 2011

Status

approved