OFFSET
1,1
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).
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
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {6049, 32257, 79201}, 40] (* Vincenzo Librandi, Jan 27 2012 *)
PROG
(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
(PARI) for(n=1, 40, print1(10368*n^2 - 4896*n + 577", ")); \\ Vincenzo Librandi, Jan 27 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Feb 26 2009
STATUS
approved