A158645 a(n) = 729*n^2 + 27.

27, 756, 2943, 6588, 11691, 18252, 26271, 35748, 46683, 59076, 72927, 88236, 105003, 123228, 142911, 164052, 186651, 210708, 236223, 263196, 291627, 321516, 352863, 385668, 419931, 455652, 492831, 531468, 571563, 613116, 656127, 700596, 746523, 793908, 842751

Offset

0,1

Comments

The identity (54*n^2 + 1)^2 - (729*n^2 + 27)*(2*n)^2 = 1 can be written as A158646(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.: -27*(1 + 25*x + 28*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 19 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(27))*Pi/sqrt(27) + 1)/54.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(27))*Pi/sqrt(27) + 1)/54. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {27, 756, 2943}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[27, 756, 2943]; [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 17 2012
(PARI) for(n=0, 40, print1(729*n^2 + 27", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158646.

Sequence in context: A279652 A284039 A159668 * A348634 A232951 A138979

Adjacent sequences: A158642 A158643 A158644 * A158646 A158647 A158648

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

Status

approved