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

1,1

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).

References

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

Links

Table of n, a(n) for n=1..39.

Example

Triples begin:
  3, 4, 5;
  7, 24, 25;
  15, 112, 113;
  31, 480, 481;

Mathematica

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

Crossrefs

Cf. A000225 (short leg), A092440 (hypotenuse), A378395, A365577, A365578, A365796

Sequence in context: A216433 A101761 A035359 * A365577 A325410 A269719

Adjacent sequences: A378960 A378961 A378962 * A378964 A378965 A378966

Keyword

nonn,tabf

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

Status

approved