A001219 Triangular numbers of form a(a+1)(a+2).

0, 6, 120, 210, 990, 185136, 258474216

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full

References

• R. K. Guy, Unsolved Problems in Number Theory, D3.

Links

• S. P. Mohanty, Which triangular numbers are products of three consecutive integers?, Acta Mathematica Hungarica 1991, Volume 58, Issue 1-2, pp 31-36.
• Eric Weisstein's World of Mathematics, Triangular Number.

Crossrefs

Cf. A000217, A097571.

Programs

Maple

count:= 0:
for a from 0 to 637 do
  p:= a*(a+1)*(a+2);
  if issqr(1+8*p) then
    count:= count+1;
    A[count]:= p;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Jan 14 2015

Mathematica

Select[Times @@@ Partition[Range[0, 700], 3, 1], OddQ[ Sqrt[ 8 # + 1]] &] (* Harvey P. Dale, Nov 04 2017 *)

Formula

a(n) = A000217(A097571(n)). - Michel Marcus, Jan 14 2015