A158589 a(n) = 324*n^2 - 18.

306, 1278, 2898, 5166, 8082, 11646, 15858, 20718, 26226, 32382, 39186, 46638, 54738, 63486, 72882, 82926, 93618, 104958, 116946, 129582, 142866, 156798, 171378, 186606, 202482, 219006, 236178, 253998, 272466, 291582, 311346, 331758, 352818, 374526, 396882, 419886

Offset

1,1

Comments

The identity (36*n^2 - 1)^2 - (324*n^2 - 18)*(2*n)^2 = 1 can be written as A136017(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

From Vincenzo Librandi, Feb 16 2012: (Start)

G.f.: -18*x*(17 + 20*x - x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/36.

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

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

E.g.f.: 18*(exp(x)*(18*x^2 + 18*x - 1) + 1).

a(n) = 18*A157910(n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {306, 1278, 2898}, 40] (* Vincenzo Librandi, Feb 16 2012 *)
324*Range[40]^2-18 (* Harvey P. Dale, Jul 25 2019 *)

Prog

(Magma) I:=[306, 1278, 2898]; [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
(PARI) for(n=1, 40, print1(324*n^2 - 18", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A136017, A157910.

Sequence in context: A294710 A206271 A236012 * A030030 A206679 A172966

Adjacent sequences: A158586 A158587 A158588 * A158590 A158591 A158592

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

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

Status

approved