A365796 - OEIS

%I #52 Sep 26 2024 23:51:35

%S 3,4,5,5,12,13,13,84,85,85,3612,3613,3613,6526884,6526885,6526885,

%T 21300113901612,21300113901613,21300113901613,

%U 226847426110843688722000884,226847426110843688722000885,226847426110843688722000885,25729877366557343481074291996721923093306518970391612,25729877366557343481074291996721923093306518970391613,25729877366557343481074291996721923093306518970391613

%N Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its short leg 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 (first 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), (c_(n-1))^2-1)/2, ((c_(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="/A365796/a365796.pdf">Capitulo 5. Catetos</a>, El Libro de las Ternas Pitagóricas.

%e Triples begin:

%e   3, 4, 5;

%e   5, 12, 13;

%e   13, 84, 85;

%e   85, 3612, 3613;

%e ...

%t {a0,b0,c0}={3,4,5};

%t m=8;

%t f[n_]:=Module[{fn0=c0,fn1=(c0^2+1)/2},Do[{fn0,fn1}={fn1,(fn0^2+1)/2},{2n-1}];fn0];t[n_]:={f[n-1],f[n]-1,f[n]};

%t ternas={a0,b0,c0};

%t For[i=1,i<=m,i++,ternas=Join[ternas,t[i]]];

%t ternas

%Y Cf. A007018 (inradius), A000058 (lower exinradius).

%Y Cf. A053631 (long leg), A053630 (hypotenuse), A365577, A365578.

%K nonn,tabf

%O 1,1

%A _Miguel-Ángel Pérez García-Ortega_, Sep 19 2023