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A238008 Numbers n such that n*(n+3)*(n+6) is a triangular number.

%I A238008 #29 Mar 10 2014 04:23:35
%S A238008 -5,-3,0,1,10,12,22,159,639,651,2629
%N A238008 Numbers n such that n*(n+3)*(n+6) is a triangular number.
%C A238008 From _Joerg Arndt_, Feb 27 2014: (Start)
%C A238008 Using x for n, we are looking for integral points on the elliptic curve y*(y+1) ==  2 * x*(x+3)*(x+6).
%C A238008 Substituting x --> x/2 and y --> y/2 and dividing the equation by 4 we obtain the Weierstrass form y^2 + 2*y == x^3 + 18*x^2 + 72*x.
%C A238008 Running the Sage program gives the following list of points (x : y : 1):
%C A238008 [(-10 : 8 : 1), (-7 : 5 : 1), (-6 : 0 : 1), (0 : 0 : 1), (2 : 14 : 1),  (20 : 128 : 1), (24 : 160 : 1), (29 : 203 : 1), (44 : 350 : 1), (318 : 5830 : 1), (1278 : 46008 : 1),  (1302 : 47304 : 1), (5258 : 381920 : 1)].
%C A238008 Dividing all x by 2 gives
%C A238008 [-5, -7/2, -3, 0, 1, 10, 12, 29/2, 22, 159, 639, 651, 2629].
%C A238008 The integral values are the terms of this sequence.
%C A238008 (End)
%e A238008 1 is in the sequence because 1*4*7=28 is a triangular number.
%o A238008 (PARI)
%o A238008 istriang(n)=issquare(8*n+1);
%o A238008 isok(n)=istriang( n*(n+3)*(n+6) );
%o A238008 for (n=-10^6, 10^6, if ( isok(n), print1(n,", ") ) );
%o A238008 \\ _Joerg Arndt_, Feb 17 2014
%o A238008 (Sage)
%o A238008 # for the curve y^2 + b1*x*y + b3*y =  x^3 + b2*x^2 + b4*x + b5, use
%o A238008 # EllipticCurve([b1,b2, b3,b4,b5])
%o A238008 # we have       y^2 +           2*y == x^3 + 18*x^2 + 72*x +  0, so need
%o A238008 E=EllipticCurve([0, 18, 2, 72, 0])
%o A238008 E.integral_points()
%o A238008 ## _Joerg Arndt_, Feb 27 2014
%o A238008 (PARI) isok(n) = ispolygonal((n-1)*n*(n+1), 3); \\ _Michel Marcus_, Mar 05 2014
%Y A238008 Cf. A000217, A165519, A165893.
%K A238008 sign,fini,full
%O A238008 1,1
%A A238008 _Alex Ratushnyak_, Feb 16 2014
%E A238008 Added the negative terms, _Joerg Arndt_, Feb 27 2014