A380299 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the area of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 13, 84, 85, 1093, 597324, 597325, 652875133, 213122969644883844, 213122969644883845, 139142687152258502421051253, 9680343693975641657052402486887446135645084826435004, 9680343693975641657052402486887446135645084826435005

Offset

1,1

Comments

The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).

References

El Libro de las Ternas Pitagóricas, Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz y José Miguel Blanco Casado, Preprint, 2025.

Links

Table of n, a(n) for n=1..15.

Miguel-Ángel Pérez García-Ortega, El libro de las ternas pitagóricas

Formula

For n >= 1, a(3*n+1) = a(3*n-2)*a(3*n-1)+1, a(3*n-1) = (a(3*n-2)^2-1)/2, and a(3*n) = a(3*n-1)+1. - Pontus von Brömssen, Feb 04 2025

Example

Triples begin:
 3, 4, 5;
 13, 84, 85;
 1093, 597324, 597325;
 652875133, 213122969644883844, 213122969644883845;

Mathematica

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

Crossrefs

Cf. A378395, A365577, A378963, A379506.

Sequence in context: A211518 A049895 A379506 * A365017 A226117 A221173

Adjacent sequences: A380296 A380297 A380298 * A380300 A380301 A380302

Keyword

nonn,tabf,easy

Author

Miguel-Ángel Pérez García-Ortega, Jan 19 2025

Status

approved