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%I #67 Nov 22 2023 22:08:42
%S 3,4,5,9,40,41,81,3280,3281,6561,21523360,21523361,43046721,
%T 926510094425920,926510094425921,1853020188851841,
%U 1716841910146256242328924544640,1716841910146256242328924544641,3433683820292512484657849089281
%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 long leg and the hypotenuse 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 (third section).
%C (a_1, b_1, c_1) = (3,4,5) and for each n > 1:
%C (a_n, b_n, c_n) = (c_(n-1)+b_(n-1), ((c_(n-1)+b_(n-1))^2-1)/2, ((c_(n-1)+b_(n-1))^2+1)/2).
%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="/A365578/a365578.pdf">Capitulo 5. Catetos</a>, El Libro de las Ternas Pitagóricas.
%F a(n) = 3^2^(n-1), a(n+1) = (a(n)^2-1)/2, a(n+2) = a(n+1)+1 for n >= 1. - _Michal Paulovic_, Nov 12 2023
%e Triples begin:
%e 3, 4, 5;
%e 9, 40, 41;
%e 81, 3280, 3281;
%e ...
%t t[1] = {3, 4, 5}; t[n_] := t[n] = Module[{a, b}, a = Total@Rest@t[n - 1]; b = (a^2 - 1)/2; {a, b, b + 1}];
%t Flatten@Table[t[n], {n, 1, 6}]
%o (PARI) my(a=1,n);for(n=1,7,a=2*a+1;print1(a, ", ");a=(a^2-1)/2;print1(a, ", ", a+1, ", "));print1("...") \\ _Michal Paulovic_, Nov 11 2023
%Y Cf. A011764 (short legs), A365577, A365796.
%K nonn,tabf
%O 1,1
%A _Miguel-Ángel Pérez García-Ortega_, Sep 20 2023
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