A158547 a(n) = 24*n^2 + 1.

1, 25, 97, 217, 385, 601, 865, 1177, 1537, 1945, 2401, 2905, 3457, 4057, 4705, 5401, 6145, 6937, 7777, 8665, 9601, 10585, 11617, 12697, 13825, 15001, 16225, 17497, 18817, 20185, 21601, 23065, 24577, 26137, 27745, 29401, 31105, 32857, 34657, 36505, 38401, 40345

Offset: 0

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

The identity (24*n^2 + 1)^2 - (144*n^2 + 12)*(2*n)^2 = 1 can be written as a(n)^2 - A158546(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Leo Tavares, Illustration: Stellar Stars.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A008594, A158546, A227776.

Programs

Magma

I:=[1, 25, 97]; [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

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 25, 97}, 50] (* Vincenzo Librandi, Feb 14 2012 *)

PARI

for(n=0, 40, print1(24*n^2+1", ")); \\ Vincenzo Librandi, Feb 14 2012

Formula

G.f.: (1 + 22*x + 25*x^2)/(1-x)^3.

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

From Amiram Eldar, Mar 02 2023: (Start)

Sum_{n>=0} 1/a(n) = 1/2 + coth(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)).

Sum_{n>=0} (-1)^n/a(n) = 1/2 + cosech(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)). (End)

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

E.g.f.: exp(x)*(1 + 24*x + 24*x^2).

a(n) = A227776(2*n). (End)

Extensions

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