A157909 a(n) = 81*n^2 - 9.

72, 315, 720, 1287, 2016, 2907, 3960, 5175, 6552, 8091, 9792, 11655, 13680, 15867, 18216, 20727, 23400, 26235, 29232, 32391, 35712, 39195, 42840, 46647, 50616, 54747, 59040, 63495, 68112, 72891, 77832, 82935, 88200, 93627, 99216, 104967, 110880, 116955, 123192

Offset: 1

Author

Vincenzo Librandi, Mar 09 2009

Keywords

• nonn
• easy

Comments

The identity (18*n^2 - 1)^2 - (81*n^2 - 9)*(2*n)^2 = 1 can be written as A157910(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012

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).

Crossrefs

Cf. A005843, A136016, A157910.

Programs

Magma

I:=[72, 315, 720]; [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 08 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {72, 315, 720}, 50] (* Vincenzo Librandi, Feb 08 2012 *)
81*Range[40]^2-9 (* Harvey P. Dale, Oct 14 2023 *)

PARI

for(n=1, 40, print1(81*n^2 - 9", ")); \\ Vincenzo Librandi, Feb 08 2012

Formula

From Vincenzo Librandi, Feb 08 2012: (Start)

G.f.: -9*x*(8 + 11*x - x^2)/(x - 1)^3.

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

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/18 - Pi/(54*sqrt(3)).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(27*sqrt(3)) - 1/18. (End)

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

E.g.f.: 9*(exp(x)*(9*x^2 + 9*x - 1) + 1).

a(n) = 9*A136016(n). (End)