A378966 - OEIS

%I #7 Dec 22 2024 16:59:26

%S 0,546,132840,27132714,5400270960,1070181351954,211922939930520,

%T 41960773653737946,8308058686721274720,1644954930586205575554,

%U 325692811387179035829960,64485533166912548464047114,12767809924078284782564882640,2527961881127459862292727058546,500523684710829430645198931758200

%N Area 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.

%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) = (A377726(n,1) * A377726(n,2))/2.

%e For n=2, the short leg is A377726(2,1) = 13 and the long leg   so the semiperimeter is then a(2) = (13 * 84)/2 =546.

%t ar[n_]:=ar[n]= Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{ra(ra-1)(2ra-1)}];areas={};Do[areas=Join[areas,FullSimplify[ar[n]]],{n,0,16}];areas

%Y Cf. A002315, A377011, A377016, A377017, A377726, A378965.

%K nonn,easy

%O 0,2

%A _Miguel-Ángel Pérez García-Ortega_, Dec 12 2024