A157910 a(n) = 18*n^2 - 1.

17, 71, 161, 287, 449, 647, 881, 1151, 1457, 1799, 2177, 2591, 3041, 3527, 4049, 4607, 5201, 5831, 6497, 7199, 7937, 8711, 9521, 10367, 11249, 12167, 13121, 14111, 15137, 16199, 17297, 18431, 19601, 20807, 22049, 23327, 24641, 25991, 27377, 28799, 30257, 31751

Offset: 1

Author

Vincenzo Librandi, Mar 09 2009

Keywords

• nonn
• easy

Comments

The identity (18*n^2 - 1)^2 - (81*n^2 - 9)*(2*n)^2 = 1 can be written as a(n)^2 - A157909(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 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. A157909, A005843.

Programs

Magma

I:=[17, 71, 161]; [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 08 2012

Mathematica

18Range[40]^2-1  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{3, -3, 1}, {17, 71, 161}, 50] (* Vincenzo Librandi, Feb 08 2012 *)

PARI

for(n=1, 40, print1(18*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 08 2012

Formula

From Vincenzo Librandi, Feb 08 2012: (Start)

G.f.: x*(-17 - 20*x + x^2)/(x - 1)^3.

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

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/2.

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