A378963 -id:A378963 - cp's OEIS frontend

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

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

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).

			Triples begin:
  3, 4, 5;
  13, 84, 85;
  183, 16744, 16745;
  33673, 566935464, 566935465;

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

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

{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

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.

Original entry on oeis.org

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

Offset: 1

Miguel-Ángel Pérez García-Ortega, Jan 19 2025

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).

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

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

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

Cf. A378395, A365577, A378963, A379506.

{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

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

cp's OEIS Frontend

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.

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