A158729 a(n) = 1156*n^2 - 34.

1122, 4590, 10370, 18462, 28866, 41582, 56610, 73950, 93602, 115566, 139842, 166430, 195330, 226542, 260066, 295902, 334050, 374510, 417282, 462366, 509762, 559470, 611490, 665822, 722466, 781422, 842690, 906270, 972162, 1040366, 1110882, 1183710, 1258850, 1336302

Offset: 1

Author

Vincenzo Librandi, Mar 25 2009

Keywords

• nonn
• easy

Comments

The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as A158730(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, A158588, A158730.

Programs

Magma

I:=[1122, 4590, 10370]; [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 20 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {1122, 4590, 10370}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

PARI

for(n=1, 40, print1(1156*n^2 - 34", ")); \\ Vincenzo Librandi, Feb 20 2012

Formula

G.f.: 34*x*(-33 - 36*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(34))*Pi/sqrt(34))/68.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/68. (End)

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

E.g.f.: 34*(exp(x)*(34*x^2 + 34*x - 1) + 1).

a(n) = 34*A158588(n). (End)

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009