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

Author

Miguel-Ángel Pérez García-Ortega, Jan 19 2025

Keywords

• nonn
• tabf
• easy

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).

Examples

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

References

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

Links

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

Crossrefs

Cf. A378395, A365577, A378963, A379506.

Programs

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

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