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A365578
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 long leg and 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, 9, 40, 41, 81, 3280, 3281, 6561, 21523360, 21523361, 43046721, 926510094425920, 926510094425921, 1853020188851841, 1716841910146256242328924544640, 1716841910146256242328924544641, 3433683820292512484657849089281
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
Cf. A011764 (short legs), A365577, A365796.
Sequence in context: A056797 A029727 A086186 * A014463 A340015 A161983
KEYWORD
nonn,tabf
STATUS
approved