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, 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

Table of n, a(n) for n=1..19.

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

Adjacent sequences: A365575 A365576 A365577 * A365579 A365580 A365581

Keyword

nonn,tabf

Author

Miguel-Ángel Pérez García-Ortega, Sep 20 2023

Status

approved