A378963 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the short leg of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 7, 24, 25, 15, 112, 113, 31, 480, 481, 63, 1984, 1985, 127, 8064, 8065, 255, 32512, 32513, 511, 130560, 130561, 1023, 523264, 523265, 2047, 2095104, 2095105, 4095, 8384512, 8384513, 8191, 33546240, 33546241, 16383, 134201344, 134201345

Offset: 1

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 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;
  7, 24, 25;
  15, 112, 113;
  31, 480, 481;
		

References

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

Crossrefs

Cf. A000225 (short leg), A092440 (hypotenuse), A378395, A365577.

Programs

Mathematica

a=Table[2^(n+1)-1,{n,1,13}];Apply[Join,Map[{#,(#^2-1)/2,(#^2+1)/2}&,a]]