A381846 Area of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

0, 0, 6, 180, 4914, 142926, 4547796, 157355484, 5842280730, 229795151586, 9475645552620, 406294220860710, 18000809380947036, 820011973477512900, 38258534425043501640, 1822437060664227775020, 88405827105467677196970, 4358079981772447955690490, 217935769988152202470568700

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, May 06 2025

Keywords

• nonn
• easy

Examples

			For n=3, the short leg is A383615(3,1) = 3 and the long leg is A383615(3,2) = 4  so the area is then a(4) = (3 * 4)/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, A383616, A131428.

Programs

Mathematica

a=Table[(2n)!/(n!(n+1)!),{n,0,18}];Apply[Join,Map[{#(#-1)(2#-1)}&,a]]

Formula

a(n) = (A383615(n,1) * A383615(n,2))/2.

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