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A365577 Sequence of the short legs 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.

%I A365577 #76 Jul 14 2025 10:03:27
%S A365577 3,7,31,511,131071,8589934591,36893488147419103231,
%T A365577 680564733841876926926749214863536422911,
%U A365577 231584178474632390847141970017375815706539969331281128078915168015826259279871
%N A365577 Sequence of the short legs 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 A365577 See Corollary 5.1.1 of García-Ortega.
%C A365577 Next term a(10) has 155 digits, so it's too big to include in data.
%H A365577 Harvey P. Dale, <a href="/A365577/b365577.txt">Table of n, a(n) for n = 1..11</a>
%H A365577 Miguel-Ángel Pérez García-Ortega, <a href="/A365577/a365577.pdf">Capitulo 5. Catetos</a>, El Libro de las Ternas Pitagóricas.
%F A365577 (a_1, b_1, c_1) = (3,4,5) and for each n > 1:
%F A365577 (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 A365577 Triangles begin:
%e A365577   3, 4, 5;
%e A365577   7, 24, 25;
%e A365577   31, 480, 481;
%e A365577   511, 130560, 130561;
%e A365577   ...
%e A365577 This sequence is the first column.
%Y A365577 Cf. A385972 (long legs), A385973 (hypotenuses).
%K A365577 nonn
%O A365577 1,1
%A A365577 _Miguel-Ángel Pérez García-Ortega_, Sep 20 2023