A365577 Sequence of the short legs of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its short leg the sum of the legs of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 7, 31, 511, 131071, 8589934591, 36893488147419103231, 680564733841876926926749214863536422911, 231584178474632390847141970017375815706539969331281128078915168015826259279871

Offset

1,1

Comments

See Corollary 5.1.1 of García-Ortega.

Next term a(10) has 155 digits, so it's too big to include in data.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..11

Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.

Formula

(a_1, b_1, c_1) = (3,4,5) and for each n > 1:

(a_n, b_n, c_n) = (a_(n-1)+b_(n-1), ((a_(n-1)+b_(n-1))^2-1)/2, ((a_(n-1)+b_(n-1))^2+1)/2)

Example

Triangles begin:
  3, 4, 5;
  7, 24, 25;
  31, 480, 481;
  511, 130560, 130561;
  ...
This sequence is the first column.

Crossrefs

Cf. A385972 (long legs), A385973 (hypotenuses).

Sequence in context: A121810 A081475 A123212 * A213437 A070231 A263049

Adjacent sequences: A365574 A365575 A365576 * A365578 A365579 A365580

Keyword

nonn

Author

Miguel-Ángel Pérez García-Ortega, Sep 20 2023

Status

approved