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A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

%I A378395 #14 Jul 13 2025 19:35:14
%S A378395 3,4,5,11,60,61,123,7564,7565,15131,114473580,114473581,228947163,
%T A378395 26208401722874284,26208401722874285,52416803445748571,
%U A378395 1373760641735119632984407274271020,1373760641735119632984407274271021
%N A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
%C A378395 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 A378395 J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2024.
%e A378395 Triples begin:
%e A378395   3, 4, 5;
%e A378395   11, 60, 61;
%e A378395   123, 7564, 7565;
%e A378395   15131, 114473580, 11447358;
%e A378395 ...
%t A378395 {a0,b0,c0}={3,4,5};f[n_]:=Module[{fn0=2c0+1,fn1=((2c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+1,((2fn1+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<=6,i++,ternas=Join[ternas,t[i]]];ternas
%Y A378395 Cf. A102847 (short leg), A365577.
%K A378395 nonn,tabf
%O A378395 1,1
%A A378395 _Miguel-Ángel Pérez García-Ortega_, Nov 24 2024