A120210 - OEIS

%I #18 Nov 27 2015 05:44:10

%S 20,30,156,600,420,1640,3660,520,2590,7140,1224,10920,8190,20880,

%T 32580,4872,19998,5220,48620,69960,3150,41470,97656,132860,19080,

%U 76830,176820,230880,131070,12740,296480,11100,375156,52360,209950,468540,64080

%N Integer squares y from the smallest solutions of y^2 = x*(a^N - x)*(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.

%C The case x congruent to 0 mod b or b congruent to 0 mod x is frequent (e.g., A120212). Note that the triples a = 3, b = 4, c = 5 and a = 4, b = 3, c = 5 provide a different result for (x, y).

%C The natural solution is y = c * b * (c-b) and x = b * (c-b) with c hypotenuse in the triple. - _Giorgio Balzarotti_, Jul 19 2006

%D G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.

%e First primitive Pythagorean triple: 3, 4, 5.

%e Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).

%e Smallest integer solution: (x, y) = (4,20).

%e First element in the sequence: y = 20.

%p flag:=1; x:=0; # a, b, c primitive Pythagorean triple

%p while flag=1 do x:=x+1; y2:=x*(a^2-x)*(x+b^2); if (floor(sqrt(y2)))^2=y2 then print(sqrt(y2)); flag:=0; fi; od;

%Y Cf. A009003, A020884, A120211-A120213.

%K nonn

%O 1,1

%A _Giorgio Balzarotti_, _Paolo P. Lava_, Jun 10 2006