A307174 Numbers that are both triangular and icosahedral.

0, 1, 1128, 87153

Offset: 1

Author

William C. Laursen, Jul 09 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 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

Crossrefs

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

Cf. A000217, A006564, A027568, A309009.

Programs

Mathematica

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

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

Extensions

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