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A383616 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

%I A383616 #12 May 11 2025 18:33:58
%S A383616 1,1,6,45,378,3486,34716,367653,4088370,47273226,564194436,6911528806,
%T A383616 86537776276,1103800077100,14305255952760,187980029453205,
%U A383616 2500329620300130,33615543018643410,456277454997741300,6246438363690689010,86175353769957832380,1197196443763413093780,16738118900201817535560
%N A383616 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
%D A383616 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.
%H A383616 Miguel-Ángel Pérez García-Ortega, <a href="/A383616/a383616.pdf">El Libro de las Ternas Pitagóricas</a>
%F A383616 a(n) = (A383615(n,1) + A383615(n,2) + A383615(n,3)) / 2.
%F A383616 a(n) = ((2n)! / (n!*(n+1)!)) * (2*(2n)! / (n!*(n+1)!) - 1).
%e A383616 For n=3, the short leg is A383615(3,1) = 3, the long leg is A383615(3,2) = 4 and the hypotenuse is A383615(3,3) = 5 so the semiperimeter is then a(3) = (3 + 4 + 5)/2 = 6.
%t A383616 a=Table[(2n)!/(n!(n+1)!),{n,0,22}];Apply[Join,Map[{#(2#-1)}&,a]]
%Y A383616 Cf. A000108, A383615, A381846, A131428.
%K A383616 nonn,easy
%O A383616 0,3
%A A383616 _Miguel-Ángel Pérez García-Ortega_, May 02 2025