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A365577
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 legs of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.
3
3, 4, 5, 7, 24, 25, 31, 480, 481, 511, 130560, 130561, 131071, 8589803520, 8589803521, 8589934591, 36893488138829168640, 36893488138829168641, 36893488147419103231, 680564733841876926889855726716117319680, 680564733841876926889855726716117319681, 680564733841876926926749214863536422911
OFFSET
1,1
COMMENTS
See Corolario 5.1.1. of the reference file (second section).
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_1, b_1, c_1) = (3,4,5) and for each n > 1:
(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)
EXAMPLE
Triples begin
3, 4, 5;
7, 24, 25;
31, 480, 481;
511, 130560, 130561;
...
MATHEMATICA
{a0, b0, c0}={3, 4, 5};
m=8;
f[n_]:=Module[{f0={a0, b0, c0}, f1={a0+b0, ((a0+b0)^2-1)/2, ((a0+b0)^2+1)/2}}, Do[{f0, f1}={f1, {Extract[f1, 1]+Extract[f1, 2], ((Extract[f1, 1]+Extract[f1, 2])^2-1)/2, ((Extract[f1, 1]+Extract[f1, 2])^2+1)/2}}, {n}]; f0]; t[n_]:={Extract[f[n], 1], Extract[f[n], 2], Extract[f[n], 3]};
ternas={a0, b0, c0}; For[i=1, i<=m, i++, ternas=Join[ternas, t[i]]];
ternas
CROSSREFS
Sequence in context: A216433 A101761 A035359 * A325410 A269719 A214626
KEYWORD
nonn,tabf,more
STATUS
approved