A158549 a(n) = 26*n^2 + 1.

1, 27, 105, 235, 417, 651, 937, 1275, 1665, 2107, 2601, 3147, 3745, 4395, 5097, 5851, 6657, 7515, 8425, 9387, 10401, 11467, 12585, 13755, 14977, 16251, 17577, 18955, 20385, 21867, 23401, 24987, 26625, 28315, 30057, 31851, 33697, 35595, 37545, 39547, 41601, 43707

Offset

0,2

Comments

The identity (26*n^2 + 1)^2 - (169*n^2 + 13)*(2*n)^2 = 1 can be written as a(n)^2 - A158548(n)*A005843(n)^2 = 1.

Links

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

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: (1 + 24*x + 27*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(26))*Pi/sqrt(26) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(26))*Pi/sqrt(26) + 1)/2. (End)

Mathematica

Table[26n^2+1, {n, 0, 50}]  (* Harvey P. Dale, Feb 21 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 27, 105}, 50] (* Vincenzo Librandi, Feb 14 2012 *)

Prog

(Magma) I:=[1, 27, 105]; [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 14 2012
(PARI) for(n=1, 40, print1(26*n^2+1", ")); \\ Vincenzo Librandi, Feb 14 2012

Crossrefs

Cf. A005843, A158548.

Sequence in context: A036346 A140376 A046347 * A254319 A044278 A044659

Adjacent sequences: A158546 A158547 A158548 * A158550 A158551 A158552

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 21 2009

Extensions

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

Status

approved