A365796 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its short leg the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 5, 12, 13, 13, 84, 85, 85, 3612, 3613, 3613, 6526884, 6526885, 6526885, 21300113901612, 21300113901613, 21300113901613, 226847426110843688722000884, 226847426110843688722000885, 226847426110843688722000885, 25729877366557343481074291996721923093306518970391612

Offset

1,1

Comments

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

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

(a_n, b_n, c_n) = (c_(n-1), (c_(n-1))^2-1)/2, ((c_(n-1))^2+1)/2).

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..23.

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

Example

Triples begin:
  3, 4, 5;
  5, 12, 13;
  13, 84, 85;
  85, 3612, 3613;
...

Mathematica

{a0, b0, c0}={3, 4, 5};
m=8;
f[n_]:=Module[{fn0=c0, fn1=(c0^2+1)/2}, Do[{fn0, fn1}={fn1, (fn0^2+1)/2}, {2n-1}]; fn0]; t[n_]:={f[n-1], f[n]-1, f[n]};
ternas={a0, b0, c0};
For[i=1, i<=m, i++, ternas=Join[ternas, t[i]]];
ternas

Crossrefs

Cf. A007018 (inradius), A000058 (lower exinradius).

Cf. A053631 (long leg), A053630 (hypotenuse), A365577, A365578.

Sequence in context: A103606 A139794 A369493 * A202819 A185383 A004484

Adjacent sequences: A365793 A365794 A365795 * A365797 A365798 A365799

Keyword

nonn,tabf,more

Author

Miguel-Ángel Pérez García-Ortega, Sep 19 2023

Status

approved