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A309009
Numbers that are both triangular and octahedral.
1
OFFSET
1,3
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
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 *)
PROG
(PARI) lista(nn) = for (n=0, nn, if (ispolygonal(k=n*(2*n^2 + 1)/3, 3), print1(k, ", "))); \\ Michel Marcus, Jul 06 2019
CROSSREFS
Intersection of A000217 and A005900. - A.H.M. Smeets, Aug 19 2019
Sequence in context: A282736 A277293 A177043 * A117064 A112001 A286314
KEYWORD
nonn,fini,full
AUTHOR
William C. Laursen, Jul 05 2019
EXTENSIONS
Keyword "full" added by Max Alekseyev, Feb 12 2024
STATUS
approved