OFFSET
1,1
COMMENTS
The identity (6n^2 - 1)^2 - (9n^2 - 3)*(2n)^2 = 1 can be written as A140811(n-1)^2 - a(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1. [broken link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: -3*x*(2 + 5*x - x^2)/(x - 1)^3. - Vincenzo Librandi, Feb 05 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 05 2012
a(n) = a(n-1) + 18*n - 9. - Vincenzo Librandi, Feb 05 2012
From Amiram Eldar, May 28 2022: (Start)
a(n) = 3*A080663(n).
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(3)*cot(Pi/sqrt(3))))/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/6. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {6, 33, 78}, 50] (* Vincenzo Librandi, Feb 05 2012 *)
9*Range[50]^2-3 (* Harvey P. Dale, Aug 04 2024 *)
PROG
(Magma) I:=[6, 33, 78]; [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 05 2012
(PARI) for(n=1, 40, print1(9*n^2 - 3", ")); \\ Vincenzo Librandi, Feb 05 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 08 2009
STATUS
approved