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Numbers n such that 2*n^2+1 is a perfect power.
1

%I #14 Feb 08 2015 04:35:57

%S 0,2,11,12,70,408,2378,13860,80782,470832,2744210,15994428,93222358,

%T 543339720,3166815962,18457556052,107578520350,627013566048,

%U 3654502875938,21300003689580,124145519261542,723573111879672,4217293152016490,24580185800219268

%N Numbers n such that 2*n^2+1 is a perfect power.

%C The value of y in the solution of the Diophantine equation x^a - 2*y^b = 1. All solutions have b=2. Sequence A075114 gives n^2. The only known solution for a>2 is y=11. See A075114 for more details.

%F Conjecture: a(n) = 6*a(n-1) - a(n-2) for n>5; g.f.: x^2*(2-x-52*x^2+9*x^3)/ (1-6*x+x^2). - _Colin Barker_, Apr 28 2012

%o (PARI)

%o Vec(x^2*(2-x-52*x^2+9*x^3)/ (1-6*x+x^2) + O(x^66))

%o /* _Joerg Arndt_, Apr 28 2012, using _Colin Barker_'s g.f. */

%K nonn

%O 1,2

%A _T. D. Noe_, Mar 29 2006

%E More terms from _T. D. Noe_, Nov 19 2006