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A365796 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. 3
3, 4, 5, 5, 12, 13, 13, 84, 85, 85, 3612, 3613, 3613, 6526884, 6526885, 6526885, 21300113901612, 21300113901613, 21300113901613, 226847426110843688722000884, 226847426110843688722000885, 226847426110843688722000885, 25729877366557343481074291996721923093306518970391612 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See Corolario 5.1.1. of the reference file (first section).
(a_1, b_1, c_1) = (3,4,5) and for each n > 1:
(a_n, b_n, c_n) = (c_(n-1), (c_(n-1))^2-1)/2, ((c_(n-1))^2+1)/2).
REFERENCES
J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
LINKS
Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.
EXAMPLE
Triples begin:
3, 4, 5;
5, 12, 13;
13, 84, 85;
85, 3612, 3613;
...
MATHEMATICA
{a0, b0, c0}={3, 4, 5};
m=8;
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]};
ternas={a0, b0, c0};
For[i=1, i<=m, i++, ternas=Join[ternas, t[i]]];
ternas
CROSSREFS
Cf. A007018 (inradius), A000058 (lower exinradius).
Cf. A053631 (long leg), A053630 (hypotenuse), A365577, A365578.
Sequence in context: A103606 A139794 A369493 * A202819 A185383 A004484
KEYWORD
nonn,tabf,more
AUTHOR
STATUS
approved

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Last modified May 26 16:43 EDT 2024. Contains 372840 sequences. (Running on oeis4.)