A158638 a(n) = 48*n^2 + 1.

1, 49, 193, 433, 769, 1201, 1729, 2353, 3073, 3889, 4801, 5809, 6913, 8113, 9409, 10801, 12289, 13873, 15553, 17329, 19201, 21169, 23233, 25393, 27649, 30001, 32449, 34993, 37633, 40369, 43201, 46129, 49153, 52273, 55489, 58801, 62209, 65713, 69313, 73009, 76801

Offset: 0

Author

Vincenzo Librandi, Mar 23 2009

Keywords

• nonn
• easy

Comments

The identity (48*n^2 + 1)^2 - (576*n^2 + 24)*(2*n)^2 = 1 can be written as a(n)^2 - A158637(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).

Crossrefs

Cf. A000290, A005843, A158480, A158637.

Programs

Magma

I:=[1, 49, 193]; [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 17 2012

Maple

A158638:=n->48*n^2 + 1; seq(A158638(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 49, 193}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
48*Range[0,40]^2+1 (* Harvey P. Dale, Mar 19 2013 *)

PARI

for(n=0, 40, print1(48*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Formula

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

G.f.: -(1 + 46*x + 49*x^2)/(x-1)^3.

a(n) = 48*A000290(n) + 1. - Wesley Ivan Hurt, Dec 06 2013

From Amiram Eldar, Mar 19 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2. (End)

From Elmo R. Oliveira, Jan 17 2025: (Start)

E.g.f.: exp(x)*(1 + 48*x + 48*x^2).

a(n) = A158480(2*n). (End)

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009