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 A380823 #29 Mar 21 2025 02:24:05
%S A380823 15,6,28,45,120,276,703,1770,4560,11781,30628,79800,208335,544446,
%T A380823 1423828,3725085,9748320,25514796,66787903,174835650,457697640,
%U A380823 1198222581,3136914028,8212428720,21500225295,56288009526,147363418828,385801624845,1010040449160,2644318093956,6922911197503
%N A380823 Semiperimeter of the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
%D A380823 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.
%H A380823 Miguel-Ángel Pérez García-Ortega, <a href="/A380823/a380823_1.pdf">El Libro de las Ternas Pitagóricas</a>.
%H A380823 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-6,4,2,-1).
%F A380823 a(n) = (A380821(n,1) + A380821(n,2) + A380821(n,3))/2.
%F A380823 a(n) = (Lucas(n) + 1)*(2*Lucas(n) + 1).
%F A380823 G.f.: (15 - 54*x + 34*x^2 + 35*x^3 - 28*x^4)/((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). - _Stefano Spezia_, Mar 08 2025
%e A380823 For n=2, the short leg is A380821(2,1) = 7, the long leg is A380821(2,2) = 24 and the hypotenuse is A380821(2,3) = 25 so the semiperimeter is then a(2) = (7 + 24 + 25)/2 = 28.
%t A380823 a=Table[LucasL[n],{n,0,30}];Apply[Join,Map[{(#+1)(2#+1)}&,a]]
%Y A380823 Cf. A380821, A380824, A381721, A000032.
%K A380823 nonn,easy
%O A380823 0,1
%A A380823 _Miguel-Ángel Pérez García-Ortega_, Feb 04 2025