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%I #26 Feb 12 2024 15:18:22
%S 0,1,6,231
%N Numbers that are both triangular and octahedral.
%C 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.
%C 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
%t triQ[n_] := IntegerQ[Sqrt[8n + 1]]; oct[n_] := n(2n^2 + 1)/3 ; Select[oct@Range[0, 10], triQ] (* _Amiram Eldar_, Jul 15 2019 *)
%o (PARI) lista(nn) = for (n=0, nn, if (ispolygonal(k=n*(2*n^2 + 1)/3, 3), print1(k, ", "))); \\ _Michel Marcus_, Jul 06 2019
%Y Intersection of A000217 and A005900. - _A.H.M. Smeets_, Aug 19 2019
%Y Cf. A000217, A005900, A027568, A307174.
%K nonn,fini,full
%O 1,3
%A _William C. Laursen_, Jul 05 2019
%E Keyword "full" added by _Max Alekseyev_, Feb 12 2024
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