A165519 Integers k for which k(k+1)(k+2) is a triangular number.

-2, -1, 0, 1, 4, 5, 9, 56, 636

Offset

1,1

Comments

This sequence is complete; there are no other integers k for which k(k+1)(k+2) is a triangular number.

Integers k such that 8*k*(k+1)*(k+2)+1 is a square. - Robert Israel, Nov 07 2014

References

R. K. Guy, "Figurate Numbers", D3 in Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, 1994, p. 148.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.

Links

Table of n, a(n) for n=1..9.

S. P. Mohanty, Which triangular numbers are products of three consecutive integers?, Acta Math Hung 58, 31-36 (1991).

Example

The third triangular number which is a product of three consecutive integers is 4*5*6=120=T(15), but 4 is the fifth integer k for which k(k+1)(k+2) is a triangular number, so a(5)=4.

Maple

select(x -> issqr(8*x^3 + 24*x^2 + 16*x+1), [$-2..1000]); # Robert Israel, Nov 07 2014

Mathematica

TriangularNumberQ[k_]:=If[IntegerQ[1/2 (Sqrt[1+8k]-1)], True, False]; Select[Range[750], TriangularNumberQ[ # (#+1)(#+2)] &]
With[{nos=Partition[Range[0, 1000], 3, 1]}, Transpose[Select[nos, IntegerQ[ (Sqrt[1+8Times@@#]-1)/2]&]][[1]]] (* Harvey P. Dale, Dec 25 2011 *)

Prog

(PARI) isok(k) = ispolygonal(k*(k+1)*(k+2), 3); \\ Michel Marcus, Oct 31 2014
(Magma) [-2, -1] cat [n: n in [0..1000] | IsSquare(8*n^3+24*n^2 +16*n+1)]; // Vincenzo Librandi, Nov 10 2014

Crossrefs

Cf. A000217, A001219.

Sequence in context: A336703 A323174 A295683 * A266972 A339650 A266493

Adjacent sequences: A165516 A165517 A165518 * A165520 A165521 A165522

Keyword

sign,fini,full

Author

Ant King, Sep 28 2009

Extensions

Initial 0 added by Alexander R. Povolotsky, Sep 29 2009

Initial -2 and -1 added by Alex Ratushnyak, Nov 07 2014

Status

approved