A158590 a(n) = 324*n^2 + 18.

18, 342, 1314, 2934, 5202, 8118, 11682, 15894, 20754, 26262, 32418, 39222, 46674, 54774, 63522, 72918, 82962, 93654, 104994, 116982, 129618, 142902, 156834, 171414, 186642, 202518, 219042, 236214, 254034, 272502, 291618, 311382, 331794, 352854, 374562, 396918

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 A158591(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).

Crossrefs

Cf. A005843, A158591.

Programs

Magma

I:=[18, 342, 1314]; [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 16 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {18, 342, 1314}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
324 Range[0,40]^2+18 (* Harvey P. Dale, Nov 22 2018 *)

PARI

for(n=0, 40, print1(324*n^2 + 18", ")); \\ Vincenzo Librandi, Feb 16 2012

Formula

G.f.: -18*(1 + 16*x + 19*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) + 1)/36.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) + 1)/36. (End)

E.g.f.: 18*exp(x)*(1 + 18*x + 18*x^2). - Elmo R. Oliveira, Jan 15 2025

Extensions

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