A379506 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the semiperimeter 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, 183, 16744, 16745, 33673, 566935464, 566935465, 1133904603, 642869824352293804, 642869824352293805, 1285739649838492213, 826563223583404284483387832630818684, 826563223583404284483387832630818685

Offset: 1

Author

Miguel-Ángel Pérez García-Ortega, Dec 23 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;
  13, 84, 85;
  183, 16744, 16745;
  33673, 566935464, 566935465;
		

Links

• Miguel-Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz, and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint, 2024.

Crossrefs

Cf. A002065 (short leg), A378395, A365577, A378963

Programs

Mathematica

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