A379509 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

7, 127, 3527, 115199, 3886471, 131868799, 4478743367, 152140105727, 5168253960967, 175568314524799, 5964153390518471, 202605640846963199, 6882627599758753927, 233806732543181952127, 7942546277657462785607, 269812766700752532479999, 9165691521506791484696071, 311363698964290393026435199

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

Keywords

• nonn
• easy

Comments

For all n: a(n) == 7 (mod 8).

Examples

			For n=2, the short leg is A377725(2,1) = 15 the long leg is A377725(2,2) = 112 so the semiperimeter is then a(2) = 15 + 112 = 127.
		

References

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

Links

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

Crossrefs

Cf. A002315, A377025, A378386, A378380.

Programs

Mathematica

s[n_]:=s[n]=Module[{r},r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{2r^2+4r+1}];sumas={};Do[semis=Join[sumas,FullSimplify[s[n]]],{n,0,17}];sumas

Formula

a(n) = A377725(n,1) + A377725(n,2).