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%I #32 Jan 29 2023 04:06:50
%S 3,5,21,15,7,55,45,39,119,77,33,65,35,9,105,91,105,297,187,95,207,117,
%T 57,377,299,217,697,459,175,319,165,51,275,209,115,403,273,85,133,63,
%U 11,171,153,203,555,345,189,429,247,155,987,777,539,1755,1161,429
%N 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 A*t, B*t, 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).
%C The tree P runs uniquely through every primitive Pythagorean triple.
%C The ternary tree is built as follows:
%C - for any n and k such that n > 0 and 0 < k <= 3^(n-1):
%C - P(n, k) is a column vector,
%C - P(n+1, 3*k-2) = A * P(n, k),
%C - P(n+1, 3*k-1) = B * P(n, k),
%C - P(n+1, 3*k) = C * P(n, k).
%C All terms are odd.
%C Every primitive Pythagorean triple (a, b, c) can be characterized by a pair of parameters (i, j) such that:
%C - i > j > 0 and gcd(i, j) = 1 and i and j are of opposite parity,
%C - a = i^2 - j^2,
%C - b = 2 * i * j,
%C - c = i^2 + j^2,
%C - 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)).
%C Every primitive Pythagorean triple (a, b, c) can also be characterized by a pair of parameters (u, v) such that:
%C - u > v > 0 and gcd(u, v) = 1 and u and v are odd,
%C - a = u * v,
%C - b = (u^2 - v^2) / 2,
%C - c = (u^2 + v^2) / 2,
%C - 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)).
%H Rémy Sigrist, <a href="/A321768/b321768.txt">Rows n = 1..9, flattened</a>
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/triples/index.html">Trees of Primitive Pythagorean Triples</a>, section UAD Tree "row-wise A leg".
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples">Tree of primitive Pythagorean triples</a>
%H <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>
%F Empirically:
%F - T(n, 1) = 2*n + 1,
%F - T(n, (3^(n-1) + 1)/2) = A046727(n),
%F - T(n, 3^(n-1)) = 4*n^2 - 1.
%e The first rows are:
%e 3
%e 5, 21, 15
%e 7, 55, 45, 39, 119, 77, 33, 65, 35
%o (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]];
%o 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])
%Y See A321769 and A321770 for the other components.
%Y See A322170 for the corresponding areas.
%Y See A322181 for the corresponding perimeters.
%Y Cf. A321782, A321783, A321784, A321785.
%Y Cf. A046727.
%K nonn,tabf
%O 1,1
%A _Rémy Sigrist_, Nov 18 2018
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