A377726 Lengths of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

84, 3280, 113764, 3878112, 131820084, 4478459440, 152138450884, 5168244315840, 175568258308884, 5964153062868112, 202605638937276964, 6882627588628286880, 233806732478308836084, 7942546277279354556400, 269812766698548756220804, 9165691521493946935370112

Offset: 1

Author

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

Keywords

• nonn
• easy

Examples

			Triangles begins:
  n=1:     13,        84,        85;
  n=2:     81,      3280,      3281;
  n=3:    477,    113764,    113765;
  ...
This sequence gives the middle column.
		

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, A377725, A362545 (short legs).

Programs

Mathematica

ra[n_]:=ra[n]=Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{2ra-1,2ra^2-2ra,2ra^2-2ra+1}];exradio={};Do[exradio=Join[exradio,FullSimplify[ra[n]]],{n,0,10}];exradio

Formula

a(n) = 2 * A002315(n) * (A002315(n) - 1).