A120210 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.

20, 30, 156, 600, 420, 1640, 3660, 520, 2590, 7140, 1224, 10920, 8190, 20880, 32580, 4872, 19998, 5220, 48620, 69960, 3150, 41470, 97656, 132860, 19080, 76830, 176820, 230880, 131070, 12740, 296480, 11100, 375156, 52360, 209950, 468540, 64080

Offset

1,1

Comments

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).

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

References

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

Links

Table of n, a(n) for n=1..37.

Example

First primitive Pythagorean triple: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Smallest integer solution: (x, y) = (4,20).
First element in the sequence: y = 20.

Maple

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

Crossrefs

Cf. A009003, A020884, A120211-A120213.

Sequence in context: A359002 A066214 A285494 * A181639 A166631 A167360

Adjacent sequences: A120207 A120208 A120209 * A120211 A120212 A120213

Keyword

nonn

Author

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

Status

approved