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A225870 Nonnegative integers of the form x*y*z*(x+y-z) with integers x>=y>=z.

%I A225870 #10 May 21 2013 15:07:25
%S A225870 0,1,4,9,12,16,24,25,36,40,45,49,60,64,72,81,84,100,105,112,120,121,
%T A225870 144,160,169,180,189,192,196,216,220,225,240,252,256,264,280,289,297,
%U A225870 300,312,324,336,352,360,361,364,384,385,396,400,420,429,432,441,480
%N A225870 Nonnegative integers of the form x*y*z*(x+y-z) with integers x>=y>=z.
%C A225870 For n>=0 and n = x*y*z*(x+y-z) with integers x>=y>=z then we can even find nonnegative solutions (x,y,z). However, if we restrict to z>=0 then there are no solutions (x,y,z) in case n<0.
%C A225870 The negative integers of the form x*y*z*(x+y-z) with integers x>=y>=z are the negatives of A213158 and in that case z<0.
%C A225870 Nonnegative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c.
%C A225870 Note that we must allow c<0 to represent n=12, 24, 40, ....
%C A225870 The negative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c are the negatives of A213158.
%e A225870 12 = (1)*(-2)*(-3)*((1)+(-2)-(-3)) with (x,y,z) = (1,-2,-3).
%e A225870 12 = 2*2*1*(2+2-1) with (x,y,z) = (2,2,1).
%e A225870 12 = ((0)^2-(-2)^2)*((-1)^2-(-2)^2) with (a,b,c) = (0,-1,-2).
%e A225870 12 = ((1)^2-(-2)^2)*((0)^2-(-2)^2) with (a,b,c) = (1,0,-2).
%o A225870 (PARI) {isa(n) = forvec( v = vector(3, i, [0, ceil(n^(1/2))]), if( n == v[1] * v[2] * v[3] * (v[3] + v[2] - v[1]), return(1)), 1)}
%Y A225870 Cf. A213158.
%K A225870 nonn
%O A225870 1,3
%A A225870 _Michael Somos_, May 18 2013