A158586 a(n) = 34*n^2 + 1.

1, 35, 137, 307, 545, 851, 1225, 1667, 2177, 2755, 3401, 4115, 4897, 5747, 6665, 7651, 8705, 9827, 11017, 12275, 13601, 14995, 16457, 17987, 19585, 21251, 22985, 24787, 26657, 28595, 30601, 32675, 34817, 37027, 39305, 41651, 44065, 46547, 49097, 51715, 54401, 57155

Offset

0,2

Comments

The identity (34*n^2 + 1)^2 - (289*n^2 + 17)*(2*n)^2 = 1 can be written as a(n)^2 - A158585(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Lisa Carbone and Pranav Shankar, Kac-Moody Fibonacci sequences, arXiv:2601.00958 [math.NT], 2026. See p. 13, Table 7.

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 + 32*x + 35*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(34))*Pi/sqrt(34) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(34))*Pi/sqrt(34) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 35, 137}, 50] (* Vincenzo Librandi, Feb 15 2012 *)

Prog

(Magma) I:=[1, 35, 137]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012
(PARI) for(n=0, 50, print1(34*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 15 2012

Crossrefs

Cf. A005843, A158585.

Sequence in context: A044367 A044748 A171473 * A350205 A267565 A330230

Adjacent sequences: A158583 A158584 A158585 * A158587 A158588 A158589

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

Status

approved