A378965 Semiperimeter 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.

1, 91, 3321, 114003, 3879505, 131828203, 4478506761, 152138726691, 5168245923361, 175568267678203, 5964153117476505, 202605639255558003, 6882627590483364721, 233806732489121022091, 7942546277342372594601, 269812766698916052264003, 9165691521496087693591105, 311363698964228006760021403

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, the long leg is A377725(2,2) = 842 and the hypotenuse is A377725(2,3) = 85 so the semiperimeter is then a(2) = (13 + 84 + 85)/2 = 91.
		

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

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

Formula

a(n) = (A377726(n,1) + A377726(n,2) + A377726(n,3))/2.