Search: id:A140811
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%I A140811
%S A140811 1,5,23,53,95,149,215,293,383,485,599,725,863,1013,1175,1349,1535,1733,
%T A140811 1943,2165,2399,2645,2903,3173,3455,3749,4055,4373,4703,5045,5399,5765,
%U A140811 6143,6533,6935,7349,7775,8213,8663,9125,9599,10085,10583,11093,11615
%V A140811 -1,5,23,53,95,149,215,293,383,485,599,725,863,1013,1175,1349,1535,1733,
               1943,2165,2399,
%W A140811 2645,2903,3173,3455,3749,4055,4373,4703,5045,5399,5765,6143,6533,6935,
               7349,7775,8213,
%X A140811 8663,9125,9599,10085,10583,11093,11615
%N A140811 6n^2 - 1.
%C A140811 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).
%C A140811 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]
%D A140811 P. Curtz, Integration .. Centre de Calcul Scientifique de l' Armement,
               Arcueil, (1969) 28-36.
%H A140811 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">
               X^2-AY^2=1</a> [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%H A140811 Edward Everett Withford, <a href="http://quod.lib.umich.edu/cgi/t/text/
               text-idx?c=umhistmath;cc=umhistmath;idno=abv2773.0001.001;view=toc">
               Pell Equation</a> [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%F A140811 a(n)=2a(n-1)-a(n-2)+12.
%F A140811 First differences: a(n+1)-a(n)=A017593(n). Second differences A071593(n+1)-A071593(n)=12.
%F A140811 G.f.: (1-8*x-5*x^2)/(x-1)^3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Aug 30 2009]
%Y A140811 A157872, A005843 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%Y A140811 Sequence in context: A055626 A127200 A147113 this_sequence A090686 A082277 
               A155851
%Y A140811 Adjacent sequences: A140808 A140809 A140810 this_sequence A140812 A140813 
               A140814
%K A140811 sign
%O A140811 0,2
%A A140811 Paul Curtz (bpcrtz(AT)free.fr), Jul 16 2008
%E A140811 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 
               06 2008
%E A140811 Better description Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 
               03 2009

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