A321784 -id:A321784 - cp's OEIS frontend

A321768 Consider the ternary tree of triples P(n, k) with n > 0 and 0 < k <= 3^(n-1), such that P(1, 1) = [3; 4; 5] and each triple t on some row branches to the triples At, Bt, C*t on the next row (with A = [1, -2, 2; 2, -1, 2; 2, -2, 3], B = [1, 2, 2; 2, 1, 2; 2, 2, 3] and C = [-1, 2, 2; -2, 1, 2; -2, 2, 3]); T(n, k) is the first component of P(n, k).

3, 5, 21, 15, 7, 55, 45, 39, 119, 77, 33, 65, 35, 9, 105, 91, 105, 297, 187, 95, 207, 117, 57, 377, 299, 217, 697, 459, 175, 319, 165, 51, 275, 209, 115, 403, 273, 85, 133, 63, 11, 171, 153, 203, 555, 345, 189, 429, 247, 155, 987, 777, 539, 1755, 1161, 429

Offset: 1

Rémy Sigrist, Nov 18 2018

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,
- A321784(n, k) and A321785(n, k) respectively give the value of u and of v pertaining to (A321768(n, k), A321769(n, k), A321770(n, k)).

			The first rows are:
   3
   5, 21, 15
   7, 55, 45, 39, 119, 77, 33, 65, 35

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
Index entries related to Pythagorean Triples

See A321769 and A321770 for the other components.
See A322170 for the corresponding areas.
See A322181 for the corresponding perimeters.
Cf. A321782, A321783, A321784, A321785.
Cf. A046727.

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])

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.

A321785 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt(A321769(n, k)^2 - A321768(n, k)^2).

Original entry on oeis.org

1, 1, 3, 3, 1, 5, 5, 3, 7, 7, 3, 5, 5, 1, 7, 7, 5, 11, 11, 5, 9, 9, 3, 13, 13, 7, 17, 17, 7, 11, 11, 3, 11, 11, 5, 13, 13, 5, 7, 7, 1, 9, 9, 7, 15, 15, 7, 13, 13, 5, 21, 21, 11, 27, 27, 11, 17, 17, 5, 19, 19, 9, 23, 23, 9, 13, 13, 3, 19, 19, 13, 29, 29, 13, 23

Offset: 1

Rémy Sigrist, Nov 22 2018

This sequence and A321784 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

			The first rows are:
   1
   1, 3, 3
   1, 5, 5, 3, 7, 7, 3, 5, 5

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

Cf. A001333, A321768, A321769, A321784.

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 (sqrtint(t[3, 1] - t[2, 1]))

Empirically:
- T(n, 1) = 1,
- T(n, (3^(n-1) + 1)/2) = A001333(n),
- T(n, 3^(n-1)) = 2*n - 1.

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