A362218 Three-column array read by rows: row n gives the unique ordered primitive Pythagorean triple (a,b,c) with a

Original entry on oeis.org

3, 4, 5, 8, 15, 17, 5, 12, 13, 12, 35, 37, 7, 24, 25, 16, 63, 65, 9, 40, 41, 20, 99, 101, 11, 60, 61, 24, 143, 145, 13, 84, 85, 28, 195, 197, 15, 112, 113, 32, 255, 257, 17, 144, 145, 36, 323, 325, 19, 180, 181, 40, 399, 401

Offset: 3

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Author

Miguel-Ángel Pérez García-Ortega, Apr 11 2023

Keywords

• nonn
• tabf

Comments

Given an ordered primitive Pythagorean triple (a,b,c) with a

For n>=3 there exists a unique ordered primitive Pythagorean triple such that (b+c)/a = n.

For n odd, the triple is {n, (n^2-1)/2, (n^2+1)/2}.

For n even, the triple is { 2*n, n^2-1, n^2+1 }.

Examples

			Irregular array begins:
  n=3:   3,  4,  5;
  n=4:   8, 15, 17;
  n=5:   5, 12, 13;
  n=6:  12, 35, 37;
  n=7:   7, 24, 25;
  ...
Row n=3 is (3,4,5) and has (b+c)/a = (4+5)/3 = 3.
Row n=4 is (8,15,17) and has (b+c)/a = (15+17)/8 = 4.
		

References

• J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

Links

• Miguel-Ángel Pérez García-Ortega, Radial proportions (in Spanish).

Crossrefs

Cf. A022998 (short leg), A066830 (long leg), A228564 (hypotenuse).

Programs

Mathematica

k=50;
ternas={{n," ",a,b,c," ",r," "," γ2 "," ",s," ",rb}};Do[If[Mod[t,2]==0,ternas=Join[ternas,{{t," ",2t,t^2-1,t^2+1," ",t-1," ",t," ",t(t+1)," ",t(t-1)}}],ternas=Join[ternas,{{t," ",t,(t^2-1)/2,(t^2+1)/2," ",(t-1)/2," ",t," ",(t(t+1))/2," ",(t(t-1))/2}}]],{t,3,k+2}]
MatrixForm[Transpose[ternas]]

Formula

T(n,1) = A022998(n).

T(n,2) = A066830(n).

T(n,3) = A228564(n).

a(6*k-3) = 2*k+1;

a(6*k-2) = ((2*k+1)^2 - 1)/2;

a(6*k-1) = ((2*k+1)^2 + 1)/2;

a(6*k) = 4*(k+1);

a(6*k+1) = 4*(k+1)^2 - 1;

a(6*k+2) = 4*(k+1)^2 + 1.

Extensions

Edited by N. J. A. Sloane, Apr 30 2023

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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