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

Author

Vincenzo Librandi, Mar 22 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A005843, A158590, A247792.

Programs

Magma

[36*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013

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 *)

PARI

a(n)=36*n^2+1 \\ Charles R Greathouse IV, Oct 16 2015

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)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: exp(x)*(1 + 36*x + 36*x^2).

a(n) = A247792(2*n). (End)

Extensions

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