OFFSET
1,1
COMMENTS
The tree P runs uniquely through every primitive Pythagorean triple.
The ternary tree is built as follows:
- for any n and k such that n > 0 and 0 < k <= 3^(n-1):
- P(n, k) is a column vector,
- P(n+1, 3*k-2) = A * P(n, k),
- P(n+1, 3*k-1) = B * P(n, k),
- P(n+1, 3*k) = C * P(n, k).
All terms are odd.
Every primitive Pythagorean triple (a, b, c) can be characterized by a pair of parameters (i, j) such that:
- i > j > 0 and gcd(i, j) = 1 and i and j are of opposite parity,
- a = i^2 - j^2,
- b = 2 * i * j,
- c = i^2 + j^2,
- A321782(n, k) and A321783(n, k) respectively give the value of i and of j pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).
Every primitive Pythagorean triple (a, b, c) can also be characterized by a pair of parameters (u, v) such that:
- u > v > 0 and gcd(u, v) = 1 and u and v are odd,
- a = u * v,
- b = (u^2 - v^2) / 2,
- c = (u^2 + v^2) / 2,
LINKS
Rémy Sigrist, Rows n = 1..9, flattened
Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree "row-wise A leg".
Wikipedia, Tree of primitive Pythagorean triples
FORMULA
Empirically:
- T(n, 1) = 2*n + 1,
- T(n, (3^(n-1) + 1)/2) = A046727(n),
- T(n, 3^(n-1)) = 4*n^2 - 1.
EXAMPLE
The first rows are:
3
5, 21, 15
7, 55, 45, 39, 119, 77, 33, 65, 35
PROG
(PARI) M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1])
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, Nov 18 2018
STATUS
approved