0,2

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).

If A=[A157872] 9*n.^2-3 (6, 33, 78, 141,.,); Y=[A005843] 2*n (except the first term , 2,4,6,8,.,); X=[A140811] 6*n^2-1 (except the first term, 5,23,5395,.,), we have, for all terms, Pell's equation X^2-A*Y^2=1. Example: 5^2-6 *2^2=1; 23^2-33*4^2=1; 53^2-78*6^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

P. Curtz, Integration .. Centre de Calcul Scientifique de l' Armement,Arcueil, (1969) 28-36.

Vincenzo Librandi, X^2-AY^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

Edward Everett Withford, Pell Equation [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

a(n)=2a(n-1)-a(n-2)+12.

First differences: a(n+1)-a(n)=A017593(n). Second differences A071593(n+1)-A071593(n)=12.

G.f.: (1-8*x-5*x^2)/(x-1)^3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Aug 30 2009]

A157872, A005843 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

Sequence in context: A055626 A127200 A147113 this_sequence A090686 A082277 A155851

Adjacent sequences: A140808 A140809 A140810 this_sequence A140812 A140813 A140814

sign

Paul Curtz (bpcrtz(AT)free.fr), Jul 16 2008

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 06 2008

Better description Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 03 2009