A158739 a(n) = 1296*n^2 + 36.

36, 1332, 5220, 11700, 20772, 32436, 46692, 63540, 82980, 105012, 129636, 156852, 186660, 219060, 254052, 291636, 331812, 374580, 419940, 467892, 518436, 571572, 627300, 685620, 746532, 810036, 876132, 944820, 1016100, 1089972, 1166436, 1245492, 1327140, 1411380

Offset

0,1

Comments

The identity (72*n^2 + 1)^2 - (1296*n^2 + 36)*(2*n)^2 = 1 can be written as A158740(n)^2 - a(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.: -36*(1+34*x+37*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>=0} 1/a(n) = (coth(Pi/6)*Pi/6 + 1)/72.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/6)*Pi/6 + 1)/72. (End)

From Elmo R. Oliveira, Jan 26 2025: (Start)

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

a(n) = 36*A158591(n). (End)

Maple

A158739:=n->1296*n^2+36: seq(A158739(n), n=0..40); # Wesley Ivan Hurt, Nov 20 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {36, 1332, 5220}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[36, 1332, 5220]; [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 21 2012
(PARI) for(n=0, 40, print1(1296*n^2 + 36", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158591, A158740.

Sequence in context: A283729 A203333 A226772 * A218518 A099366 A095657

Adjacent sequences: A158736 A158737 A158738 * A158740 A158741 A158742

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved