OFFSET
1,1
COMMENTS
See Corolario 5.1.1. of the reference file (third section).
(a_1, b_1, c_1) = (3,4,5) and for each n > 1:
(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).
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.
FORMULA
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
EXAMPLE
Triples begin:
3, 4, 5;
9, 40, 41;
81, 3280, 3281;
...
MATHEMATICA
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}];
Flatten@Table[t[n], {n, 1, 6}]
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Miguel-Ángel Pérez García-Ortega, Sep 20 2023
STATUS
approved