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%I #29 Dec 11 2024 19:21:43
%S 6,120,3486,114960,3885078,131860680,4478696046,152139829920,
%T 5168252353446,175568305155480,5964153335910078,202605640528682160,
%U 6882627597903676086,233806732532369766120,7942546277594444747406,269812766700385236436800,9165691521504650726475078,311363698964277915773152440
%N Semiperimeter of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
%D Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%F a(n) = (A377725(n,1) + A377725(n,2) + A377725(n,3))/2.
%e For n=2, the short leg is A377725(2,1) = 15, the long leg is A377725(2,2) = 112 and the hypotenuse is A377725(2,3) = 113 so the semiperimeter is then a(2) = (15 + 112 + 113)/2 = 120.
%t s[n_]:=s[n]=Module[{r},r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{(r+1)(2r+1)}];semis={};Do[semis=Join[semis,FullSimplify[s[n]]],{n,0,17}];semis
%Y Cf. A002315, A377025, A378386
%K nonn,easy,new
%O 0,1
%A _Miguel-Ángel Pérez García-Ortega_, Nov 24 2024