A216269 Numbers k such that k^2 - 1 is a tetrahedral number (A000292).

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

Offset

1,2

Comments

Corresponding tetrahedral numbers are in A216268.

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

The above equation can be transformed to Y^2 = X^3 + 18*X^2 + 72*X + 1296 by setting X = 6*y and Y = 36*x, and then we can use for example SageMath to compute the integral points on this elliptic curve. - Jianing Song, Jan 09 2026

Links

Table of n, a(n) for n=1..7.

Wikipedia, Siegel's theorem on integral points.

Mathematica

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 *)

Prog

(Python)
import math
for i in range(1<<30):
    t = i*(i+1)*(i+2)//6 + 1
    sr = math.isqrt(t)
    if sr*sr == t:
        print(sr)
(SageMath)
E = EllipticCurve([0, 18, 0, 72, 1296])
pts = E.integral_points()
set(-P[1]//36 for P in pts if (P[1] != 0) & (P[1] % 36 == 0)) # Jianing Song, Jan 09 2026

Crossrefs

Cf. A000292, A216268.

Cf. A003556 (both square and tetrahedral).

Sequence in context: A270691 A270280 A270727 * A094555 A271056 A271255

Adjacent sequences: A216266 A216267 A216268 * A216270 A216271 A216272

Keyword

nonn,fini,full

Author

Alex Ratushnyak, Mar 15 2013

Status

approved