A158636 a(n) = 576*n^2 - 24.

552, 2280, 5160, 9192, 14376, 20712, 28200, 36840, 46632, 57576, 69672, 82920, 97320, 112872, 129576, 147432, 166440, 186600, 207912, 230376, 253992, 278760, 304680, 331752, 359976, 389352, 419880, 451560, 484392, 518376, 553512, 589800, 627240, 665832, 705576

Offset: 1

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 A065532(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..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. A005843, A065532, A158544.

Programs

Magma

I:=[552, 2280, 5160]; [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

Mathematica

LinearRecurrence[{3, -3, 1}, {552, 2280, 5160}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

PARI

for(n=1, 40, print1(576*n^2 - 24", ")); \\ Vincenzo Librandi, Feb 17 2012

Formula

G.f.: 24*x*(-23 - 26*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 19 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)))/48.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)) - 1)/48. (End)

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

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

a(n) = 24*A158544(n). (End)

Extensions

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