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

Author

Miguel-Ángel Pérez García-Ortega, May 02 2025

Keywords

• nonn
• easy

Examples

			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.
		

References

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

Links

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

Crossrefs

Cf. A000108, A383615, A381846, A131428.

Programs

Mathematica

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

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).