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 A377016 #26 Jul 13 2025 17:33:53
%S A377016 1,28,861,28680,970921,32963140,1119662181,38034888528,1292062686481,
%T A377016 43892073946540,1491038320325421,50651410052600280,
%U A377016 1720656899012149561,58451683130389395028,1985636569382856677301,67453191675004485098400,2291422880375627492063521,77840924741066359629967420
%N A377016 Semiperimeter of the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
%C A377016 a(0) = 1 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.
%D A377016 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A377016 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (41,-246,246,-41,1).
%F A377016 a(n) = (A002315(n) + 2*A008844(n) - 1)/2.
%F A377016 G.f.: (1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)). - _Andrew Howroyd_, Oct 14 2024
%e A377016 For n=2, the short leg is A002315(2) = 41 and the hypotenuse is A008844(n) = 841 so the semiperimeter is then a(2) = (41 + 840 + 841)/2 = 861.
%t A377016 s[n_]:=s[n]=Module[{a, b},a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;b=(a^2-1)/2;{(a+2b+1)/2}];semis={};Do[semis=Join[semis,FullSimplify[s[n]]],{n,0,17}];semis
%o A377016 (PARI) Vec((1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)) + O(x^20)) \\ _Andrew Howroyd_, Oct 14 2024
%Y A377016 Cf. A002315, A078522, A008844.
%K A377016 nonn,easy
%O A377016 0,2
%A A377016 _Miguel-Ángel Pérez García-Ortega_, Oct 13 2024