A158737 a(n) = 1296*n^2 - 36.

1260, 5148, 11628, 20700, 32364, 46620, 63468, 82908, 104940, 129564, 156780, 186588, 218988, 253980, 291564, 331740, 374508, 419868, 467820, 518364, 571500, 627228, 685548, 746460, 809964, 876060, 944748, 1016028, 1089900, 1166364, 1245420, 1327068, 1411308, 1498140

Offset

1,1

Comments

The identity (72*n^2 - 1)^2 - (1296*n^2 - 36)*(2*n)^2 = 1 can be written as A158738(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.: 36*x*(-35 - 38*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/6)*Pi/6)/72 = (1 - Pi/(2*sqrt(3)))/72.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/6)*Pi/6 - 1)/72. (End)

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

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

a(n) = 36*A136017(n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1260, 5148, 11628}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[1260, 5148, 11628]; [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 20 2012
(PARI) for(n=1, 40, print1(1296*n^2 - 36", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A136017, A158738.

Sequence in context: A350375 A179690 A350039 * A252277 A203402 A047634

Adjacent sequences: A158734 A158735 A158736 * A158738 A158739 A158740

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved