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

0,3

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..18.

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

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).

Example

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.

Mathematica

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

Crossrefs

Cf. A000108, A383615, A383616, A131428.

Sequence in context: A046989 A210358 A368730 * A135395 A384029 A337756

Adjacent sequences: A381843 A381844 A381845 * A381847 A381848 A381849

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, May 06 2025

Status

approved