A307174 Numbers that are both triangular and icosahedral.

0, 1, 1128, 87153

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 of 0 and 1.

Terms n*(n+1)/2 = m*(5*m^2-5*m+2)/2 corresponds to integral points (X,Y) = (5*m,5*n) on the elliptic curve Y^2 + Y = X^3 - 5*X^2 + 10*X, which can be computed efficiently. There are none besides those already listed. - Max Alekseyev, Feb 12 2024

Links

Table of n, a(n) for n=1..4.

Mathematica

Intersection[Accumulate[Range[0, 199]], Table[n (5n^2 - 5n + 2)/2, {n, 0, 99}]] (* Alonso del Arte, Jul 10 2019 *)

Prog

(PARI) lista(nn) = for (n=0, nn, if (ispolygonal(k=n*(5*n^2 - 5*n + 2)/2, 3), print1(k, ", "))); \\ Michel Marcus, Jul 10 2019

Crossrefs

Intersection of A000217 and A006564. - A.H.M. Smeets, Aug 19 2019

Cf. A000217, A006564, A027568, A309009.

Sequence in context: A252299 A203063 A237294 * A278732 A278434 A198469

Adjacent sequences: A307171 A307172 A307173 * A307175 A307176 A307177

Keyword

nonn,fini,full

Author

William C. Laursen, Jul 09 2019

Extensions

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

Status

approved