OFFSET
1,1
COMMENTS
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.: 27*x*(-26 - 29*x + x^2)/(x-1)^3
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 19 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(27))*Pi/sqrt(27))/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(27))*Pi/sqrt(27) - 1)/54. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {702, 2889, 6534}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
729*Range[40]^2-27 (* Harvey P. Dale, Jan 22 2014 *)
PROG
(Magma) I:=[702, 2889, 6534]; [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 17 2012
(PARI) for(n=1, 40, print1(729*n^2 - 27", ")); \\ Vincenzo Librandi, Feb 17 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 23 2009
EXTENSIONS
Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
STATUS
approved