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%I #55 Nov 22 2023 22:08:18
%S 3,4,5,7,24,25,31,480,481,511,130560,130561,131071,8589803520,
%T 8589803521,8589934591,36893488138829168640,36893488138829168641,
%U 36893488147419103231,680564733841876926889855726716117319680,680564733841876926889855726716117319681,680564733841876926926749214863536422911
%N Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its short leg the sum of the legs of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
%C See Corolario 5.1.1. of the reference file (second section).
%D J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
%H Miguel-Ángel Pérez García-Ortega, <a href="/A365577/a365577.pdf">Capitulo 5. Catetos</a>, El Libro de las Ternas Pitagóricas.
%F (a_1, b_1, c_1) = (3,4,5) and for each n > 1:
%F (a_n, b_n, c_n) = (a_(n-1)+b_(n-1), ((a_(n-1)+b_(n-1))^2-1)/2, ((a_(n-1)+b_(n-1))^2+1)/2)
%e Triples begin
%e 3, 4, 5;
%e 7, 24, 25;
%e 31, 480, 481;
%e 511, 130560, 130561;
%e ...
%t {a0,b0,c0}={3,4,5};
%t m=8;
%t f[n_]:=Module[{f0={a0,b0,c0},f1={a0+b0,((a0+b0)^2-1)/2,((a0+b0)^2+1)/2}},Do[{f0,f1}={f1,{Extract[f1,1]+Extract[f1,2],((Extract[f1,1]+Extract[f1,2])^2-1)/2,((Extract[f1,1]+Extract[f1,2])^2+1)/2}},{n}];f0];t[n_]:={Extract[f[n],1],Extract[f[n],2],Extract[f[n],3]};
%t ternas={a0,b0,c0};For[i=1,i<=m,i++,ternas=Join[ternas,t[i]]];
%t ternas
%Y Cf. A365578, A365796.
%K nonn,tabf,more
%O 1,1
%A _Miguel-Ángel Pérez García-Ortega_, Sep 20 2023