A158443 a(n) = 16*n^2 - 4.

12, 60, 140, 252, 396, 572, 780, 1020, 1292, 1596, 1932, 2300, 2700, 3132, 3596, 4092, 4620, 5180, 5772, 6396, 7052, 7740, 8460, 9212, 9996, 10812, 11660, 12540, 13452, 14396, 15372, 16380, 17420, 18492, 19596, 20732, 21900, 23100, 24332, 25596, 26892, 28220, 29580

Offset: 1

Author

Vincenzo Librandi, Mar 19 2009

Keywords

• nonn
• easy

Comments

The identity (8*n^2 - 1)^2 - (16*n^2 - 4) *(2*n)^2 = 1 can be written as A157914(n)^2 - a(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 12, in the direction 12, 60, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

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. A000466, A005843, A074377, A157914.

Programs

Magma

I:=[12, 60, 140]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Mathematica

16Range[60]^2-4  (* Harvey P. Dale, Mar 18 2011 *)

PARI

a(n) = 16*n^2 - 4.

Formula

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

G.f: 4*x*(3+6*x-x^2)/(1-x)^3.

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/16. (End)

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

E.g.f.: 4*(exp(x)*(4*x^2 + 4*x - 1) + 1).

a(n) = 4*A000466(n). (End)