A157872 a(n) = 9*n^2 - 3.

6, 33, 78, 141, 222, 321, 438, 573, 726, 897, 1086, 1293, 1518, 1761, 2022, 2301, 2598, 2913, 3246, 3597, 3966, 4353, 4758, 5181, 5622, 6081, 6558, 7053, 7566, 8097, 8646, 9213, 9798, 10401, 11022, 11661, 12318, 12993, 13686, 14397, 15126, 15873, 16638, 17421, 18222

Offset: 1

Author

Vincenzo Librandi, Mar 08 2009

Keywords

• nonn
• easy

Comments

The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as A140811(n-1)^2 - a(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1. [broken link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A080663, A140811.

Programs

Magma

I:=[6, 33, 78]; [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 05 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {6, 33, 78}, 50] (* Vincenzo Librandi, Feb 05 2012 *)
9*Range[50]^2-3 (* Harvey P. Dale, Aug 04 2024 *)

PARI

for(n=1, 40, print1(9*n^2 - 3", ")); \\ Vincenzo Librandi, Feb 05 2012

Formula

From Vincenzo Librandi, Feb 05 2012: (Start)

G.f.: -3*x*(2 + 5*x - x^2)/(x - 1)^3.

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

a(n) = a(n-1) + 18*n - 9. (End)

From Amiram Eldar, May 28 2022: (Start)

a(n) = 3*A080663(n).

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

Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/6. (End)

E.g.f.: 3*(exp(x)*(3*x^2 + 3*x - 1) + 1). - Elmo R. Oliveira, Jan 25 2025