

A152005


Numbers whose square is the product of two distinct tetrahedral numbers A000292.


0



2, 140, 280, 1092, 166460, 189070, 665840, 804540, 845460, 34250920, 38336088, 133784560, 138535992, 225792840, 4998790160, 6301258040, 7559616818, 8367691640, 39991371446, 104637102152, 227490888350, 1497809326860, 296523233581822
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OFFSET

1,1


COMMENTS

There may be values that are not given in the recurrence shown. This sequence is suggested by Ulas, p.11, who supplied the recurrence. Abstract: In this paper we give solutions of certain Diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians  Number Theory, Krakow, 1420 September 2008.
a(24) > 3*10^14.  Donovan Johnson, Jan 11 2012


LINKS

Table of n, a(n) for n=1..23.
Maciej Ulas, On certain Diophantine equations related to triangular and tetrahedral numbers, Nov 15, 2008.


FORMULA

a(n) = T(i)*T(j) where T(k) = A000292(k) = C(k+2,3) = k*(k+1)*(k+2)/6. If we define sequences u(n), v(n) recursively in the following way: u(0) = 1, v(0) = 1, u(n) = 17*u(n1) + 36*v(n1) + 20, v(n) = 8*u(n1) + 17*v(n1) + 10, then for each natural number n the following identity holds (v(n)*u(n)*(u(n + 1)))^2 = T(u(n))*T(2*u(n)). But this does not exclude the possibility that there are other elements of a(n) that do not come from this recurrence.


EXAMPLE

From R. J. Mathar, Jan 22 2009: (Start)
2 is in the sequence because 2^2 = 4*1 = T(2)*T(1).
140 is in the sequence 140^2 = 560*35 = T(14)*T(5) = 19600*1 = T(48)*T(1).
280 is in the sequence because 280^2 = 19600*4 = T(48)*T(2).
1092 is in the sequence because 1092^2 = 3276*364 = T(26)*T(12). (End)


MATHEMATICA

(* This program is not suitable to compute more than a dozen terms. *)
terms = 12; imin = 1; imax = 3000;
Union[Reap[Do[k2 = i(i+1)(i+2)/6 j(j+1)(j+2)/6; k = Sqrt[k2]; If[IntegerQ[k], Print[k]; Sow[k]], {i, imin, imax}, {j, i+1, imax}]][[2, 1]]][[1 ;; terms]] (* JeanFrançois Alcover, Oct 31 2018 *)


CROSSREFS

Cf. A000292.
Sequence in context: A265176 A101232 A093887 * A140898 A220513 A120814
Adjacent sequences: A152002 A152003 A152004 * A152006 A152007 A152008


KEYWORD

nonn,more


AUTHOR

Jonathan Vos Post, Nov 19 2008


EXTENSIONS

Sequence replaced by sequence with no intermediate terms missing by R. J. Mathar, Jan 22 2009
a(15)a(18) from Donovan Johnson, Jan 24 2009
a(19)a(23) from Donovan Johnson, Jan 11 2012


STATUS

approved



