A120097 -id:A120097 - 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.

A094193 Values y of the generator pairs (x, y), x>y of primitive Pythagorean triples, sorted on x.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 5, 2, 4, 6, 1, 3, 5, 7, 2, 4, 8, 1, 3, 7, 9, 2, 4, 6, 8, 10, 1, 5, 7, 11, 2, 4, 6, 8, 10, 12, 1, 3, 5, 9, 11, 13, 2, 4, 8, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 16, 1, 5, 7, 11, 13, 17, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 3, 7, 9, 11, 13, 17, 19, 2, 4, 8, 10, 16

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

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 riples

Cf. A094192, A094194, A120097, A120098.

A120086 Numerators of expansion of Debye function for n=4: D(4,x).

Original entry on oeis.org

1, -2, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120087.

See the W. Lang link under A120080 for more details on the general case D(n,x), n= 1, 2, ... D(4,x) is the e.g.f. of the rational sequence {4*B(n)/(n+4)}, n >= 0. See A227573/A227574. - Wolfdieter Lang, Jul 17 2013

			Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=1, with a factor (x^4)/4 extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A060054. [From R. J. Mathar, Aug 07 2008]
Cf. A000367/A002445, A027641/A027642, A120097, A227573/A227574 (D(4,x) as e.g.f.). - Wolfdieter Lang, Jul 17 2013
Cf. A120080, A120081, A120082, A120083, A120084, A120085, A120087 (denominators).

[Numerator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // G. C. Greubel, May 02 2023

r[n_]:= 4*BernoulliB[n]/((n+4)*n!); Table[r[n]//Numerator, {n,0,36}] (* Jean-François Alcover, Jun 21 2013 *)

[numerator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = numerator(r(n)), with r(n) = [x^n](1 - 4*x/(2*(4+1)) + 2*Sum_{k >= 0} (B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = numerator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - Wolfdieter Lang, Jul 17 2013

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

A120098 Short leg of primitive Pythagorean triangle corresponding to hypotenuse A094194.

Original entry on oeis.org

3, 5, 8, 7, 20, 9, 12, 11, 28, 33, 13, 16, 48, 39, 15, 36, 65, 17, 20, 60, 51, 19, 44, 88, 85, 57, 21, 24, 119, 95, 23, 52, 104, 133, 105, 69, 25, 28, 84, 140, 115, 75, 27, 60, 120, 161, 29, 32, 96, 160, 207, 175, 135, 87, 31, 68, 136, 204, 225, 189, 145, 93, 33, 36

Offset: 1

Lekraj Beedassy, Jun 08 2006

nonn

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

Cf. A120097.

cp's OEIS Frontend

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

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A094193 Values y of the generator pairs (x, y), x>y of primitive Pythagorean triples, sorted on x.

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A120086 Numerators of expansion of Debye function for n=4: D(4,x).

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

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A120098 Short leg of primitive Pythagorean triangle corresponding to hypotenuse A094194.

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