A102344 - OEIS

%I #37 Jan 01 2024 11:30:06

%S 2,7,97,1351,18817,262087,3650401,50843527,708158977,9863382151,

%T 137379191137,1913445293767,26650854921601,371198523608647,

%U 5170128475599457,72010600134783751,1002978273411373057,13969685227624439047,194572614913330773601,2710046923559006391367

%N Numbers n such that the Diophantine equation (x+2)^3-x^3=2*n^2 has solutions.

%C n^2 = 3*(2*x+4)^2+16.

%C Essentially the same as A011943. - _Chris Boyd_, Apr 18 2015

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).

%F a(n+2) = 14*a(n+1)-a(n) for n>=2.

%F G.f.: x*(2-21*x+x^2)/(1-14*x+x^2). a(n)=7*A007655(n+2)-97*A007655(n+1), n>1. - _R. J. Mathar_, Sep 11 2008

%e The first examples are 2^3-0^3=2*2^2 ; 5^3-3^3=2*7^2 ; 57^3-55^3=2*97^2 ; 781^3-779^3=2*1351^2 ; 10865^3-10863^3=2*18817^2

%p 2, seq(othopoly[T](n,7),n=1..50); # _Robert Israel_, Apr 19 2015

%t a[1]=2; a[2]=7; a[3]=97; a[n_] := a[n] = 14*a[n-1]-a[n-2]; Table[a[n], {n, 1, 17}] (* _Jean-François Alcover_, Dec 17 2013 *)

%t LinearRecurrence[{14,-1},{2,7,97},20] (* _Harvey P. Dale_, Sep 26 2016 *)

%o (Magma) I:=[2,7,97]; [n le 3 select I[n] else 14*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Apr 19 2015

%o (PARI) Vec(x*(2-21*x+x^2)/(1-14*x+x^2) + O(x^30)) \\ _Michel Marcus_, Apr 19 2015

%Y Cf. A011943.

%K easy,nonn

%O 1,1

%A _Richard Choulet_, Sep 08 2008

%E More terms from _Vincenzo Librandi_, Apr 19 2015