A383616 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

1, 1, 6, 45, 378, 3486, 34716, 367653, 4088370, 47273226, 564194436, 6911528806, 86537776276, 1103800077100, 14305255952760, 187980029453205, 2500329620300130, 33615543018643410, 456277454997741300, 6246438363690689010, 86175353769957832380, 1197196443763413093780, 16738118900201817535560

Offset

0,3

References

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

Links

Table of n, a(n) for n=0..22.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Formula

a(n) = (A383615(n,1) + A383615(n,2) + A383615(n,3)) / 2.

a(n) = ((2n)! / (n!*(n+1)!)) * (2*(2n)! / (n!*(n+1)!) - 1).

Example

For n=3, the short leg is A383615(3,1) = 3, the long leg is A383615(3,2) = 4 and the hypotenuse is A383615(3,3) = 5 so the semiperimeter is then a(3) = (3 + 4 + 5)/2 = 6.

Mathematica

a=Table[(2n)!/(n!(n+1)!), {n, 0, 22}]; Apply[Join, Map[{#(2#-1)}&, a]]

Crossrefs

Cf. A000108, A383615, A381846, A131428.

Sequence in context: A007194 A025551 A101600 * A233668 A243694 A227169

Adjacent sequences: A383613 A383614 A383615 * A383617 A383618 A383619

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, May 02 2025

Status

approved