A378966 Area of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 546, 132840, 27132714, 5400270960, 1070181351954, 211922939930520, 41960773653737946, 8308058686721274720, 1644954930586205575554, 325692811387179035829960, 64485533166912548464047114, 12767809924078284782564882640, 2527961881127459862292727058546, 500523684710829430645198931758200

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

Keywords

• nonn
• easy

Examples

			For n=2, the short leg is A377726(2,1) = 13 and the long leg   so the semiperimeter is then a(2) = (13 * 84)/2 =546.
		

References

• Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Crossrefs

Cf. A002315, A377016, A377017, A377726, A378965.

Programs

Mathematica

ar[n_]:=ar[n]= Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{ra(ra-1)(2ra-1)}];areas={};Do[areas=Join[areas,FullSimplify[ar[n]]],{n,0,16}];areas

Formula

a(n) = (A377726(n,1) * A377726(n,2))/2.