A158768 a(n) = 1521*n^2 + 39.

39, 1560, 6123, 13728, 24375, 38064, 54795, 74568, 97383, 123240, 152139, 184080, 219063, 257088, 298155, 342264, 389415, 439608, 492843, 549120, 608439, 670800, 736203, 804648, 876135, 950664, 1028235, 1108848, 1192503, 1279200, 1368939, 1461720, 1557543, 1656408

Offset

0,1

Comments

The identity (78*n^2 + 1)^2 - (1521*n^2 + 39)*(2*n)^2 = 1 can be written as A158769(n)^2 - a(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.: -39*(1 + 37*x + 40*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 23 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(39))*Pi/sqrt(39) + 1)/78.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(39))*Pi/sqrt(39) + 1)/78. (End)

E.g.f.: 39*exp(x)*(1 + 39*x + 39*x^2). - Elmo R. Oliveira, Jan 26 2025

Mathematica

LinearRecurrence[{3, -3, 1}, {39, 1560, 6123}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[39, 1560, 6123]; [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(1521*n^2 + 39", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158769.

Sequence in context: A112617 A009983 A269028 * A139191 A319490 A327589

Adjacent sequences: A158765 A158766 A158767 * A158769 A158770 A158771

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

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

Status

approved