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%I A118465 #26 Feb 01 2023 19:22:27
%S A118465 0,9,66,219,516,1005,1734,2751,4104,5841,8010,10659,13836,17589,21966,
%T A118465 27015,32784,39321,46674,54891,64020,74109,85206,97359,110616,125025,
%U A118465 140634,157491,175644,195141,216030,238359,262176,287529,314466,343035
%N A118465 a(n) = 8*n^3 + n.
%C A118465 (8*n^3 + n, 8*n^3 - n) solves the Diophantine equation 2*(X-Y)^3-(X+Y)=0.
%C A118465 (m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 and where m is a positive integer. Also ((m*n^k+n)/2, (m*n^k-n)/2) solves the Diophantine equation: m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 where m is an odd number.
%C A118465 24*a(n) = (4n+1)^3 + (4n)^3 + (4n-1)^3. [_Bruno Berselli_, May 12 2014]
%H A118465 Vincenzo Librandi, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0ExMTg0NjUvYjExODQ2NS50eHQ">Table of n, a(n) for n = 0..1000</a>
%H A118465 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wNA">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A118465 G.f.: 3*x*(x+3)*(3*x+1)/(-1+x)^4. - _R. J. Mathar_, Nov 14 2007
%F A118465 a(n) = n*A081585(n). - _Vincenzo Librandi_, May 13 2014
%p A118465 A118465:=n->8*n^3 + n; seq(A118465(n), n=0..30); # _Wesley Ivan Hurt_, May 13 2014
%t A118465 Table[8 n^3 + n, {n, 0, 35}]
%t A118465 CoefficientList[Series[3 x (x + 3) (3 x + 1)/(-1 + x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 13 2014 *)
%t A118465 LinearRecurrence[{4,-6,4,-1},{0,9,66,219},40] (* _Harvey P. Dale_, Feb 01 2023 *)
%o A118465 (Magma) [8*n^3 + n: n in [0..30]]; // _Wesley Ivan Hurt_, May 13 2014
%Y A118465 Cf. A006003, A081585.
%K A118465 nonn,easy
%O A118465 0,2
%A A118465 _Mohamed Bouhamida_, May 16 2006, Oct 02 2007
%E A118465 Edited by _Stefan Steinerberger_, Jul 24 2007

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