A102344 Numbers n such that the Diophantine equation (x+2)^3-x^3=2*n^2 has solutions.

2, 7, 97, 1351, 18817, 262087, 3650401, 50843527, 708158977, 9863382151, 137379191137, 1913445293767, 26650854921601, 371198523608647, 5170128475599457, 72010600134783751, 1002978273411373057, 13969685227624439047, 194572614913330773601, 2710046923559006391367

Offset

1,1

Comments

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

Essentially the same as A011943. - Chris Boyd, Apr 18 2015

Links

Table of n, a(n) for n=1..20.

Index entries for linear recurrences with constant coefficients, signature (14,-1).

Formula

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

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

Example

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

Maple

2, seq(othopoly[T](n, 7), n=1..50); # Robert Israel, Apr 19 2015

Mathematica

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 *)
LinearRecurrence[{14, -1}, {2, 7, 97}, 20] (* Harvey P. Dale, Sep 26 2016 *)

Prog

(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
(PARI) Vec(x*(2-21*x+x^2)/(1-14*x+x^2) + O(x^30)) \\ Michel Marcus, Apr 19 2015

Crossrefs

Cf. A011943.

Sequence in context: A072059 A308961 A240696 * A087589 A002812 A219280

Adjacent sequences: A102341 A102342 A102343 * A102345 A102346 A102347

Keyword

easy,nonn

Author

Richard Choulet, Sep 08 2008

Extensions

More terms from Vincenzo Librandi, Apr 19 2015

Status

approved