%I #19 May 31 2024 14:04:01

%S 0,1,3,8,20,34

%N The indices of tetrahedral numbers that are also triangular.

%H E. T. Avanesov, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa12/aa12128.pdf">Solution of a problem on figurate numbers</a> (in Russian), Acta Arith. 12 1966/1967 pages 409-420.

%e The eighth tetrahedral number is 120, which is also a triangular number. Hence, 8 belongs to the sequence.

%t Select[Table[{n,(n(n+1)(n+2))/6},{n,0,35}],OddQ[Sqrt[1+8#[[2]]]]&][[;;,1]] (* _Harvey P. Dale_, May 31 2024 *)

%o (PARI) for(n=0,34,if(ispolygonal(n*(n+1)*(n+2)/6,3),print1(n", "))) \\ _Charles R Greathouse IV_, Apr 07 2013

%Y Cf. A027568, A102349.

%K nonn,full,fini

%O 0,3

%A _Tanya Khovanova_, Apr 06 2013