A158591 a(n) = 36*n^2 + 1.

1, 37, 145, 325, 577, 901, 1297, 1765, 2305, 2917, 3601, 4357, 5185, 6085, 7057, 8101, 9217, 10405, 11665, 12997, 14401, 15877, 17425, 19045, 20737, 22501, 24337, 26245, 28225, 30277, 32401, 34597, 36865, 39205, 41617, 44101, 46657, 49285, 51985, 54757, 57601

Offset

0,2

Comments

The identity (36*n^2+1)^2 - (324*n^2+18)*(2*n)^2 = 1 can be written as a(n)^2 - A158590(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -(1+34*x+37*x^2)/(x-1)^3.

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/6)*Pi/6 + 1)/2.

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

Mathematica

CoefficientList[Series[- (1 + 34 x + 37 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)
36*Range[0, 40]^2+1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 37, 145}, 40] (* Harvey P. Dale, Jul 02 2019 *)

Prog

(Magma) [36*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013
(PARI) a(n)=36*n^2+1 \\ Charles R Greathouse IV, Oct 16 2015

Crossrefs

Cf. A005843, A158590.

Sequence in context: A142498 A337258 A350405 * A262318 A262921 A031690

Adjacent sequences: A158588 A158589 A158590 * A158592 A158593 A158594

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved