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%I A307174 #48 Feb 12 2024 10:34:04
%S A307174 0,1,1128,87153
%N A307174 Numbers that are both triangular and icosahedral.
%C A307174 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.
%C A307174 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
%t A307174 Intersection[Accumulate[Range[0, 199]], Table[n (5n^2 - 5n + 2)/2, {n, 0, 99}]] (* _Alonso del Arte_, Jul 10 2019 *)
%o A307174 (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
%Y A307174 Intersection of A000217 and A006564. - _A.H.M. Smeets_, Aug 19 2019
%Y A307174 Cf. A000217, A006564, A027568, A309009.
%K A307174 nonn,fini,full
%O A307174 1,3
%A A307174 _William C. Laursen_, Jul 09 2019
%E A307174 Keyword "full" added by _Max Alekseyev_, Feb 12 2024

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