OFFSET
1,1
COMMENTS
The identity (42*n^2-1)^2 - (441*n^2-21)*(2*n)^2 = 1 can be written as a(n)^2-A145678(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 16 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(-41-44*x+x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(42))*Pi/sqrt(42))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(42))*Pi/sqrt(42) - 1)/2. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {41, 167, 377}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
PROG
(Magma) I:=[41, 167, 377]; [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, Feb 16 2012
(PARI) for(n=1, 40, print1(42*n^2-1", ")); \\ Vincenzo Librandi, Feb 16 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
EXTENSIONS
Edited by R. J. Mathar, Jul 26 2009
A-number updated by R. J. Mathar, Oct 16 2009
STATUS
approved