A309009 Numbers that are both triangular and octahedral.

0, 1, 6, 231

Offset: 1

Author

William C. Laursen, Jul 05 2019

Keywords

• nonn
• fini
• full

Comments

Similar to the list of triangular and tetrahedral numbers (A027568). It would appear that the similar sequence of pentagonal-dodecahedral numbers contains only the trivial cases 0 and 1.

Terms n*(n+1)/2 = m*(2*m^2+1)/3 corresponds to integral points (X,Y) = (12*m,36*n+18) on the elliptic curve Y^2 = X^3 + 72*X + 324, which can be computed efficiently. There are none besides those already listed. - Max Alekseyev, Feb 12 2024

Is 1156 the only nontrivial number that is both octahedral and square? - Kelvin Voskuijl, Jan 07 2025

Crossrefs

Intersection of A000217 and A005900. - A.H.M. Smeets, Aug 19 2019

Cf. A000217, A005900, A027568, A307174.

Programs

Mathematica

triQ[n_] := IntegerQ[Sqrt[8n + 1]]; oct[n_] := n(2n^2 + 1)/3 ; Select[oct@Range[0, 10], triQ] (* Amiram Eldar, Jul 15 2019 *)

PARI

lista(nn) = for (n=0, nn, if (ispolygonal(k=n*(2*n^2 + 1)/3, 3), print1(k, ", "))); \\ Michel Marcus, Jul 06 2019

Extensions

Keyword "full" added by Max Alekseyev, Feb 12 2024