A158556 a(n) = 28*n^2 + 1.

1, 29, 113, 253, 449, 701, 1009, 1373, 1793, 2269, 2801, 3389, 4033, 4733, 5489, 6301, 7169, 8093, 9073, 10109, 11201, 12349, 13553, 14813, 16129, 17501, 18929, 20413, 21953, 23549, 25201, 26909, 28673, 30493, 32369, 34301, 36289, 38333, 40433, 42589, 44801, 47069

Offset

0,2

Comments

The identity (28*n^2 + 1)^2 - (196*n^2 + 14) * (2*n)^2 = 1 can be written as a(n)^2 - A158555(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (1 + 26*x + 29*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 09 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(28))*Pi/sqrt(28) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(28))*Pi/sqrt(28) + 1)/2. (End)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: exp(x)*(1 + 28*x + 28*x^2).

a(n) = A247541(2*n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 29, 113}, 50] (* Vincenzo Librandi, Mar 02 2012 *)
28*Range[0, 40]^2+1 (* Harvey P. Dale, Jun 30 2022 *)

Prog

(Magma) I:=[1, 29, 113]; [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, Mar 02 2012
(PARI) for(n=0, 40, print1(28*n^2+1", ")); \\ Vincenzo Librandi, Mar 02 2012

Crossrefs

Cf. A005843, A158555, A247541.

Sequence in context: A338023 A139946 A093359 * A272629 A201602 A050527

Adjacent sequences: A158553 A158554 A158555 * A158557 A158558 A158559

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 21 2009

Extensions

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

Status

approved