A094193 -id:A094193 - cp's OEIS frontend

A094192 Values x of the generator pairs (x, y), x>y of primitive Pythagorean triples, sorted.

2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19

Offset: 1

Lekraj Beedassy, May 25 2004

nonn

The generated primitive Pythagorean triple (X, Y, Z), with XA120098, Y=A120097, Z=A094194. - Lekraj Beedassy, Jul 12 2006

Ordered A147847 (?). - Paul Curtz, Nov 16 2008

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 145.

Ray Chandler, Table of n, a(n) for n = 1..10000
F. Richman, Pythagorean Triples

Cf. A094193, A094194, A120097, A120098.

A094194 Hypotenuses x^2 + y^2 of primitive Pythagorean triangles, sorted on values x of the generator pair (x, y), x>y.

Original entry on oeis.org

5, 13, 17, 25, 29, 41, 37, 61, 53, 65, 85, 65, 73, 89, 113, 85, 97, 145, 101, 109, 149, 181, 125, 137, 157, 185, 221, 145, 169, 193, 265, 173, 185, 205, 233, 269, 313, 197, 205, 221, 277, 317, 365, 229, 241, 289, 421, 257, 265, 281, 305, 337, 377, 425, 481, 293

Offset: 1

Lekraj Beedassy, May 25 2004

nonn

For ordered hypotenuses of primitive Pythagorean triangles see A020882.

The hypotenuse Z of the primitive Pythagorean triple (X, Y, Z) with Xy (x and y coprime and not both odd) using the relation Z = x^2 + y^2. The even leg is 2*x*y and the odd leg is x^2 - y^2. [From Lekraj Beedassy, May 06 2010]

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 145.

Ray Chandler, Table of n, a(n) for n = 1..10000
F. Richman, Pythagorean Triples

Cf. A094192, A094193, A120097, A120098.

Inserted a sqrt(.) operation in the definition - R. J. Mathar, Apr 12 2010

Deleted incorrect sqrt in definition (based on author's initial comment) - Aaron Kastel, Oct 30 2012

A126611 Sum x+y of generator pairs (x, y) {x and y coprime and not both odd} of primitive Pythagorean triangles, sorted.

Original entry on oeis.org

3, 5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 27, 27, 27

Offset: 1

Lekraj Beedassy, Feb 07 2007

nonn

Also, the square root of the sum of even leg and hypotenuse of primitive Pythagorean triangles, sorted.

Cf. A094192, A094193.

2n-1 appears A072451(n) times.

A309424 Sum x+y of generator pairs (x, y) {x and y coprime and not both odd} of primitive Pythagorean triangles, sorted by x and y (for same x).

Original entry on oeis.org

3, 5, 5, 7, 7, 9, 7, 11, 9, 11, 13, 9, 11, 13, 15, 11, 13, 17, 11, 13, 17, 19, 13, 15, 17, 19, 21, 13, 17, 19, 23, 15, 17, 19, 21, 23, 25, 15, 17, 19, 23, 25, 27, 17, 19, 23, 29, 17, 19, 21, 23, 25, 27, 29, 31, 19, 21, 23, 25, 27, 29, 31, 33, 19, 23, 25, 29, 31, 35, 21, 23, 25, 27, 29, 31, 33, 35, 37, 21, 23, 27, 29, 31, 33, 37, 39

Offset: 1

Rui Lin, Jul 31 2019

nonn

This sequence is based on x and y (for same x) in increasing order, directly mapping to A094192 and A094193, while A126611 is sorted by the sum x+y.
Given any 2 of below 4 sequences, primitive Pythagorean triangles can be generated.
A094192: the bigger one in generator pairs;
A094193: the smaller one in generator pairs;
A309424: the sum of generator pairs;
A309425: the difference of generator pairs.

Cf. A094192, A094193, A126611, A309425.

a(n) = A094192(n) + A094193(n).

A309425 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x > y} of primitive Pythagorean triangles, sorted by x and y (for same x).

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 5, 1, 5, 3, 1, 7, 5, 3, 1, 7, 5, 1, 9, 7, 3, 1, 9, 7, 5, 3, 1, 11, 7, 5, 1, 11, 9, 7, 5, 3, 1, 13, 11, 9, 5, 3, 1, 13, 11, 7, 1, 15, 13, 11, 9, 7, 5, 3, 1, 15, 13, 11, 9, 7, 5, 3, 1, 17, 13, 11, 7, 5, 1, 17, 15, 13, 11, 9, 7, 5, 3, 1, 19, 17, 13, 11, 9, 7, 3, 1

Offset: 1

Rui Lin, Jul 31 2019

nonn

This sequence is based on x and y (for same x) in increasing order, directly mapping to A094192 and A094193, while A126637 is sorted by the sum x+y.
Given any two of the four sequences below, primitive Pythagorean triangles can be generated.
A094192: the bigger one in generator pairs;
A094193: the smaller one in generator pairs;
A309424: the sum of generator pairs;
A309425: the difference of generator pairs.

Cf. A094192, A094193, A126637, A309424.

a(n) = A094192(n) - A094193(n).

A126637 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x>y} of primitive Pythagorean triangles, sorted on values x+y (A126611), then on x-y.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, 1, 7, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 5, 11, 13, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 1, 5, 7, 11

Offset: 1

Lekraj Beedassy, Feb 08 2007

This sequence gives the consecutive rows n = 2*m + 1, for m >= 1, of the array A216319. See the example. - Wolfdieter Lang, Oct 24 2019

			From _Wolfdieter Lang_, Oct 24 2019: (Start)
From the array A216319 with n = 2*m + 1 = x + y, for m >= 1, the (x, y) values giving the terms of the present sequence as values x-y are:
m, n \ k    1      2      3      4      5      6 ...   x-y values
--------------------------------------------------------------------
1,  3:   (2,1)                                         1
2,  5:   (3,2) (4,1)                                   1 3
3,  7:   (4,3) (5,2)   (6,1)                           1 3  5
4,  9:   (5,4) (7,2)   (8,1)                           1 5  7
5, 11:   (6,5) (7,4)   (8,3)  (9,2) (10,1)             1 3  5  7  9
6, 13:   (7,6) (8,5)   (9,4) (10,3) (11,2) (12,1)      1 3  5  7  9  11
7, 15:   (8,7) (11,4) (13,2) (14,1)                    1 7 11 13
... (End)

Cf. A094192, A094193, A126611, A216319.

cp's OEIS Frontend

A094192 Values x of the generator pairs (x, y), x>y of primitive Pythagorean triples, sorted.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A094194 Hypotenuses x^2 + y^2 of primitive Pythagorean triangles, sorted on values x of the generator pair (x, y), x>y.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A126611 Sum x+y of generator pairs (x, y) {x and y coprime and not both odd} of primitive Pythagorean triangles, sorted.

Views

Author

Keywords

Comments

Crossrefs

Formula

A309424 Sum x+y of generator pairs (x, y) {x and y coprime and not both odd} of primitive Pythagorean triangles, sorted by x and y (for same x).

Views

Author

Keywords

Comments

Crossrefs

Formula

A309425 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x > y} of primitive Pythagorean triangles, sorted by x and y (for same x).

Views

Author

Keywords

Comments

Crossrefs

Formula

A126637 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x>y} of primitive Pythagorean triangles, sorted on values x+y (A126611), then on x-y.

Views

Author

Keywords

Comments

Examples

Crossrefs