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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A268984 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

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Last modified May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)