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A380299 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the area of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

%I A380299 #18 Jul 13 2025 19:35:57
%S A380299 3,4,5,13,84,85,1093,597324,597325,652875133,213122969644883844,
%T A380299 213122969644883845,139142687152258502421051253,
%U A380299 9680343693975641657052402486887446135645084826435004,9680343693975641657052402486887446135645084826435005
%N A380299 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the area of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
%C A380299 The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).
%D A380299 El Libro de las Ternas Pitagóricas, Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz y José Miguel Blanco Casado, Preprint, 2025.
%H A380299 Miguel-Ángel Pérez García-Ortega, <a href="/A380299/a380299.pdf">El libro de las ternas pitagóricas</a>
%F A380299 For n >= 1, a(3*n+1) = a(3*n-2)*a(3*n-1)+1, a(3*n-1) = (a(3*n-2)^2-1)/2, and a(3*n) = a(3*n-1)+1. - _Pontus von Brömssen_, Feb 04 2025
%e A380299 Triples begin:
%e A380299  3, 4, 5;
%e A380299  13, 84, 85;
%e A380299  1093, 597324, 597325;
%e A380299  652875133, 213122969644883844, 213122969644883845;
%t A380299 {a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=a0 b0+1,fn1=((a0 b0+1)^2-1)/2},Do[{fn0,fn1}={fn1 fn0+1,((fn1 fn0+1)^2-1)/2},{n}];fn0];t[n_]:= {f[n-1],(f[n-1]^2-1)/2,(f[n-1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=5,i++,ternas=Join[ternas,t[i]]];ternas
%Y A380299 Cf. A378395, A365577, A378963, A379506.
%K A380299 nonn,tabf,easy
%O A380299 1,1
%A A380299 _Miguel-Ángel Pérez García-Ortega_, Jan 19 2025