A158644 a(n) = 52*n^2 + 1.

1, 53, 209, 469, 833, 1301, 1873, 2549, 3329, 4213, 5201, 6293, 7489, 8789, 10193, 11701, 13313, 15029, 16849, 18773, 20801, 22933, 25169, 27509, 29953, 32501, 35153, 37909, 40769, 43733, 46801, 49973, 53249, 56629, 60113, 63701, 67393, 71189, 75089, 79093, 83201

Offset

0,2

Comments

The identity (52*n^2 + 1)^2 - (676*n^2 + 26)*(2*n)^2 = 1 can be written as a(n)^2 - A158643(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.: -(1 + 50*x + 53*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/(2*sqrt(13)))*Pi/(2*sqrt(13)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(13)))*Pi/(2*sqrt(13)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 53, 209}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[1, 53, 209]; [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(52*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158643.

Sequence in context: A142088 A330810 A005146 * A158656 A013536 A142000

Adjacent sequences: A158641 A158642 A158643 * A158645 A158646 A158647

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