This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378965 #9 Jul 13 2025 17:35:07
%S A378965 1,91,3321,114003,3879505,131828203,4478506761,152138726691,
%T A378965 5168245923361,175568267678203,5964153117476505,202605639255558003,
%U A378965 6882627590483364721,233806732489121022091,7942546277342372594601,269812766698916052264003,9165691521496087693591105,311363698964228006760021403
%N 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.
%D A378965 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%F A378965 a(n) = (A377726(n,1) + A377726(n,2) + A377726(n,3))/2.
%e A378965 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.
%t A378965 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
%Y A378965 Cf. A002315, A377016, A377017, A377726, A378965.
%K A378965 nonn,easy
%O A378965 0,2
%A A378965 _Miguel-Ángel Pérez García-Ortega_, Dec 12 2024