A365577 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its short leg the sum of the legs of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 7, 24, 25, 31, 480, 481, 511, 130560, 130561, 131071, 8589803520, 8589803521, 8589934591, 36893488138829168640, 36893488138829168641, 36893488147419103231, 680564733841876926889855726716117319680, 680564733841876926889855726716117319681, 680564733841876926926749214863536422911

Offset

1,1

Comments

See Corolario 5.1.1. of the reference file (second section).

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=1..22.

Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.

Formula

(a_1, b_1, c_1) = (3,4,5) and for each n > 1:

(a_n, b_n, c_n) = (a_(n-1)+b_(n-1), ((a_(n-1)+b_(n-1))^2-1)/2, ((a_(n-1)+b_(n-1))^2+1)/2)

Example

Triples begin
  3, 4, 5;
  7, 24, 25;
  31, 480, 481;
  511, 130560, 130561;
  ...

Mathematica

{a0, b0, c0}={3, 4, 5};
m=8;
f[n_]:=Module[{f0={a0, b0, c0}, f1={a0+b0, ((a0+b0)^2-1)/2, ((a0+b0)^2+1)/2}}, Do[{f0, f1}={f1, {Extract[f1, 1]+Extract[f1, 2], ((Extract[f1, 1]+Extract[f1, 2])^2-1)/2, ((Extract[f1, 1]+Extract[f1, 2])^2+1)/2}}, {n}]; f0]; t[n_]:={Extract[f[n], 1], Extract[f[n], 2], Extract[f[n], 3]};
  ternas={a0, b0, c0}; For[i=1, i<=m, i++, ternas=Join[ternas, t[i]]];
  ternas

Crossrefs

Cf. A365578, A365796.

Sequence in context: A216433 A101761 A035359 * A325410 A269719 A214626

Adjacent sequences: A365574 A365575 A365576 * A365578 A365579 A365580

Keyword

nonn,tabf,more

Author

Miguel-Ángel Pérez García-Ortega, Sep 20 2023

Status

approved