A120211 -id:A120211 - cp's OEIS frontend

A120212 "a" values providing solution x = b in A120211 (i.e., y^2 = b^2(a^2 - b)(b + 1) with a, b legs in primitive Pythagorean triangles).

3, 5, 7, 8, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75

Offset: 1

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

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

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

A120213 "a" values providing solution x != b in A120211, i.e., in y^2 = x(a^2 - x)(b^2 + x) with a, b legs in primitive Pythagorean triads.

Original entry on oeis.org

4, 12, 12, 15, 15, 16, 20, 20, 21, 24, 24, 28, 28, 32, 33, 35, 36, 36, 39, 40, 40, 44, 44, 45, 48, 48, 51, 52, 52, 55, 56, 56, 57, 60, 60, 60, 60, 63, 64, 65, 68, 68, 69, 72, 72, 75, 76, 76, 77, 77

Offset: 0

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

			Primitive Pythagorean triad: 4, 3, 5.
Weierstrass equation: y^2 = x*(4^2 - x)*(3^2 + x).
Smallest integer solution: (x, y) = (6,30).
As x != b, a = 4 is in the sequence.

Cf. A120210, A120211, A120212.

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.

Original entry on oeis.org

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

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

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

			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.

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

Cf. A009003, A020884, A120211-A120213.

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;

A120336 Number of solutions (x,y) of Diophantine equation y^2 = x(a^N - x)(b^N + x) (Weierstrass elliptic equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg "a".

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Jun 22 2006

nonn

Triads a = 3, b = 4, c = 5 and a = 4, b = 3, c = 5 provide different results for (x,y).

			First primitive Pythagorean triad: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Unique integer solution: (x,y) = (4,20).
First element in the sequence: 1.
Fifth primitive Pythagorean triad: 8, 15, 17.
Integer solutions: (x,y) = (15, 420) and (30, 510).
Fifth element in the sequence: 2.

Cf. A009003, A020884, A120210, A120211, A120212, A120213.

# a,b,c primitive Pythagorean triad n_sol:=0; for x from 1 by 1 to a^2 do y2:= x*( a^2 - x)*( x+ b^2); if ((floor(sqrt(y2)))^2=y2) n_sol:=n_sol+1;fi; print(n_sol) ; od;

cp's OEIS Frontend

A120212 "a" values providing solution x = b in A120211 (i.e., y^2 = b^2(a^2 - b)(b + 1) with a, b legs in primitive Pythagorean triangles).

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A120213 "a" values providing solution x != b in A120211, i.e., in y^2 = x(a^2 - x)(b^2 + x) with a, b legs in primitive Pythagorean triads.

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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.

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A120336 Number of solutions (x,y) of Diophantine equation y^2 = x(a^N - x)(b^N + x) (Weierstrass elliptic equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg "a".

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Maple

cp's OEIS Frontend

A120212 "a" values providing solution x = b in A120211 (i.e., y^2 = b^2*(a^2 - b)*(b + 1) with a, b legs in primitive Pythagorean triangles).

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Author

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A120213 "a" values providing solution x != b in A120211, i.e., in y^2 = x*(a^2 - x)*(b^2 + x) with a, b legs in primitive Pythagorean triads.

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Author

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Examples

Crossrefs

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.

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Maple

A120336 Number of solutions (x,y) of Diophantine equation y^2 = x*(a^N - x)*(b^N + x) (Weierstrass elliptic equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg "a".

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Author

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Comments

Examples

Crossrefs

Programs

Maple

A120212 "a" values providing solution x = b in A120211 (i.e., y^2 = b^2(a^2 - b)(b + 1) with a, b legs in primitive Pythagorean triangles).

A120213 "a" values providing solution x != b in A120211, i.e., in y^2 = x(a^2 - x)(b^2 + x) with a, b legs in primitive Pythagorean triads.

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.

A120336 Number of solutions (x,y) of Diophantine equation y^2 = x(a^N - x)(b^N + x) (Weierstrass elliptic equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg "a".