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 A377017 #34 Jul 13 2025 17:34:10
%S A377017 0,84,17220,3412920,675761016,133797385260,26491207202460,
%T A377017 5245125232676784,1038508304885968560,205619399242324129860,
%U A377017 40711602541676078766516,8060691683852625858745320,1595976241800278270688414120,315995235184771245126273789084,62565460590342906257639745449100
%N A377017 Area 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 A377017 a(0)=0 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.
%C A377017 All terms in this sequence are divisible by 84.
%D A377017 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A377017 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (204,-1190,204,-1).
%F A377017 a(n) = A002315(n)*(A008844(n)-1)/2.
%F A377017 G.f.: 84*x*(1 + x)/((1 - 198*x + x^2)*(1 - 6*x + x^2)). - _Andrew Howroyd_, Oct 14 2024
%e A377017 For n=2, the short leg is A002315(2) = 41 and the long leg is A008844(2)-1 = 840 so the area is then a(2) = 41*840/2 = 17220.
%t A377017 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*b)/2}]; areas={}; Do[areas=Join[areas, FullSimplify[s[n]]], {n, 0, 17}]; areas
%Y A377017 Cf. A377016, A002315, A078522, A008844.
%K A377017 nonn
%O A377017 0,2
%A A377017 _Miguel-Ángel Pérez García-Ortega_, Oct 13 2024