A158062 a(n) = 36*n^2 - 2*n.

34, 140, 318, 568, 890, 1284, 1750, 2288, 2898, 3580, 4334, 5160, 6058, 7028, 8070, 9184, 10370, 11628, 12958, 14360, 15834, 17380, 18998, 20688, 22450, 24284, 26190, 28168, 30218, 32340, 34534, 36800, 39138, 41548, 44030, 46584, 49210, 51908

Offset: 1

Author

Vincenzo Librandi, Mar 12 2009

Keywords

• nonn
• easy

Comments

The identity (36*n - 1)^2 - (36*n^2 - 2*n)*6^2 = 1 can be written as (A044102(n+1) - 1)^2 - a(n)*6^2 = 1. - Vincenzo Librandi, Feb 11 2012

The continued fraction expansion of sqrt(a(n)) is [6n-1; {1, 4, 1, 12n-2}]. - Magus K. Chu, Nov 08 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(6^2*t-2)).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A044102.

Programs

Magma

[36*n^2 - 2*n: n in [1..50]]

Mathematica

LinearRecurrence[{3, -3, 1}, {34, 140, 318}, 50] (* Vincenzo Librandi, Feb 11 2012 *)

PARI

for(n=1, 50, print1(36*n^2 - 2*n ", ")); \\ Vincenzo Librandi, Feb 11 2012

Formula

G.f.: x*(-34 - 38*x)/(x-1)^3. - Vincenzo Librandi, Feb 11 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 11 2012