

A165519


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


1




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

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


LINKS

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


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



