A216269 - OEIS

%I #20 Nov 14 2024 10:59:02

%S 1,6,11,45,51,209,660099

%N Numbers n such that n^2 - 1 is a tetrahedral number (A000292).

%C Corresponding tetrahedral numbers are in A216268.

%C The curve 6*(x^2-1)-y*(y+1)*(y+2)=0 is elliptic, and has finitely many integral points by Siegel's theorem. - _Robert Israel_, Apr 22 2021

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points">Siegel's theorem on integral points</a>

%t t = {}; Do[tet = n (n + 1) (n + 2)/6; If[IntegerQ[s = Sqrt[tet + 1]], AppendTo[t, s]], {n, 0, 100000}]; t (* _T. D. Noe_, Mar 18 2013 *)

%o (Python)

%o import math

%o for i in range(1<<30):

%o     t = i*(i+1)*(i+2)//6 + 1

%o     sr = int(math.sqrt(t))

%o     if sr*sr == t:

%o         print(sr)

%Y Cf. A000292, A216268.

%Y Cf. A003556 (both square and tetrahedral).

%K nonn,fini

%O 1,2

%A _Alex Ratushnyak_, Mar 15 2013