A158646 a(n) = 54*n^2 + 1.

1, 55, 217, 487, 865, 1351, 1945, 2647, 3457, 4375, 5401, 6535, 7777, 9127, 10585, 12151, 13825, 15607, 17497, 19495, 21601, 23815, 26137, 28567, 31105, 33751, 36505, 39367, 42337, 45415, 48601, 51895, 55297, 58807, 62425, 66151, 69985, 73927, 77977, 82135, 86401

Offset

0,2

Comments

The identity (54*n^2 + 1)^2 - (729*n^2 + 27)*(2*n)^2 = 1 can be written as a(n)^2 - A158645(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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.: -(1 + 52*x + 55*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>=0} 1/a(n) = (coth(Pi/(3*sqrt(6)))*Pi/(3*sqrt(6)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(3*sqrt(6)))*Pi/(3*sqrt(6)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 55, 217}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[1, 55, 217]; [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=0, 40, print1(54*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158645.

Sequence in context: A161763 A189005 A105442 * A294461 A013550 A254148

Adjacent sequences: A158643 A158644 A158645 * A158647 A158648 A158649

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