A158655 a(n) = 729*n^2 - 27.

702, 2889, 6534, 11637, 18198, 26217, 35694, 46629, 59022, 72873, 88182, 104949, 123174, 142857, 163998, 186597, 210654, 236169, 263142, 291573, 321462, 352809, 385614, 419877, 455598, 492777, 531414, 571509, 613062, 656073, 700542, 746469, 793854, 842697, 892998

Offset

1,1

Comments

The identity (54*n^2 - 1)^2 - (729*n^2 - 27)*(2*n)^2 = 1 can be written as A158656(n)^2 - a(n)*A005843(n)^2 = 1.

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

Cf. A158656, A005843.

Sequence in context: A305508 A305174 A316762 * A305071 A203350 A321641

Adjacent sequences: A158652 A158653 A158654 * A158656 A158657 A158658

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