A157267 a(n) = 10368*n^2 - 4896*n + 577.

6049, 32257, 79201, 146881, 235297, 344449, 474337, 624961, 796321, 988417, 1201249, 1434817, 1689121, 1964161, 2259937, 2576449, 2913697, 3271681, 3650401, 4049857, 4470049, 4910977, 5372641, 5855041, 6358177, 6882049, 7426657, 7992001, 8578081, 9184897, 9812449

Offset: 1

Author

Vincenzo Librandi, Feb 26 2009

Keywords

• nonn
• easy

Comments

The identity (10368*n^2 - 4896*n + 577)^2 - (36*n^2 - 17*n + 2)*(1728*n - 408)^2 = 1 can be written as a(n)^2 - A157265(n)*A157266(n)^2 = 1. - Vincenzo Librandi, Jan 27 2012

This is the case s = 4*n - 1 of the identity (2*r^2 - 1)^2 - ((r^2 - 1)/144)*(24*r)^2 = 1, where r = 18*s + 9*i^(s*(s+1)) - (-1)^s - 9 and i = sqrt(-1). - Bruno Berselli, Jan 29 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A157265, A157266.

Programs

Magma

I:=[6049, 32257, 79201]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 27 2012

Mathematica

LinearRecurrence[{3,-3,1},{6049,32257,79201},40] (* Vincenzo Librandi, Jan 27 2012 *)

PARI

for(n=1, 40, print1(10368*n^2 - 4896*n + 577", ")); \\ Vincenzo Librandi, Jan 27 2012

Formula

From Vincenzo Librandi, Jan 27 2012: (Start)

G.f.: x*(6049 + 14110*x + 577*x^2)/(1-x)^3.

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

E.g.f.: exp(x)*(10368*x^2 + 5472*x + 577) - 577. - Elmo R. Oliveira, Nov 09 2024