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 A165519 Integers k for which k(k+1)(k+2) is a triangular number. 1
 -2, -1, 0, 1, 4, 5, 9, 56, 636 (list; graph; refs; listen; history; text; internal format)
 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, Springer-Verlag, 1994, p. 148. LINKS 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

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)