A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 11, 60, 61, 123, 7564, 7565, 15131, 114473580, 114473581, 228947163, 26208401722874284, 26208401722874285, 52416803445748571, 1373760641735119632984407274271020, 1373760641735119632984407274271021

Offset: 1

Author

Miguel-Ángel Pérez García-Ortega, Nov 24 2024

Keywords

• nonn
• tabf

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;
  11, 60, 61;
  123, 7564, 7565;
  15131, 114473580, 11447358;
...
		

References

• J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2024.

Crossrefs

Cf. A102847 (short leg), A365577.

Programs

Mathematica

{a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=2c0+1,fn1=((2c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+1,((2fn1+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<=6,i++,ternas=Join[ternas,t[i]]];ternas