A362218 Three-column array read by rows: row n gives the unique ordered primitive Pythagorean triple (a,b,c) with a<b such that (b+c)/a = n.

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

3,1

Comments

Given an ordered primitive Pythagorean triple (a,b,c) with a<b, we call (b+c)/a the median radial proportion. This number gives us the proportion (r_b)/r, where r_b is the exinradius associated to the greater cathetus of the Pythagorean triangle (a,b,c) and r is the inradius of the same triangle.

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 }.

References

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

Links

Table of n, a(n) for n=3..56.

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

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.

Example

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.

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]]

Crossrefs

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

Sequence in context: A055742 A263041 A216888 * A106048 A219038 A258454

Adjacent sequences: A362215 A362216 A362217 * A362219 A362220 A362221

Keyword

nonn,tabf

Author

Miguel-Ángel Pérez García-Ortega, Apr 11 2023

Extensions

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

Status

approved