A158740 a(n) = 72*n^2 + 1.

1, 73, 289, 649, 1153, 1801, 2593, 3529, 4609, 5833, 7201, 8713, 10369, 12169, 14113, 16201, 18433, 20809, 23329, 25993, 28801, 31753, 34849, 38089, 41473, 45001, 48673, 52489, 56449, 60553, 64801, 69193, 73729, 78409, 83233, 88201, 93313, 98569, 103969, 109513

Offset

0,2

Comments

The identity (72*n^2 + 1)^2 - (1296*n^2 + 36)*(2*n)^2 = 1 can be written as a(n)^2 - A158739(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 + 70*x + 73*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 73, 289}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[1, 73, 289]; [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 21 2012
(PARI) for(n=0, 40, print1(72*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A158739, A005843.

Sequence in context: A142489 A033244 A140857 * A174334 A142614 A158744

Adjacent sequences: A158737 A158738 A158739 * A158741 A158742 A158743

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved