Golf a Numerical Growing Braid Braid Description

In this braid, when a strand crosses over the top of another strand it adds the other strand's value to itself and all other strand values pass through. The braid has three strands and each strand begins at 1. The first crossover is the leftmost strand crossing over the middle strand. The next crossover is the rightmost strand crossing over the new middle strand (previously the leftmost strand). These two steps of crossovers repeat. In other words, the first crossover is [a, b, c] -&gt; [b, a+b, c] and the second is [a, b, c] -&gt; [a, b+c, b]. Using these rules here are the first six levels of the braid:

1,1,1
1,2,1
1,3,2
3,4,2
3,6,4
6,9,4


Your task

Write a golfed program or function that accepts an integer as the braid level and outputs the three values for that level of the braid. You must indicate if your levels are zero- or one-based. Input and output may come in any reasonable format and trailing white space is allowed.

Test Cases (1-based)

1 -&gt; 1,1,1

2 -&gt; 1,2,1

5 -&gt; 3,6,4

10 -&gt; 28,41,19
MATL, 18 17 16 bytes
7Bi:&quot;2:4PB@EX!Y*

Input is 0-based.
Try it online! Or verify all test cases.
Explanation
Given a row vector [a b c], the next vector is obtained post-matrix-multiplying it by either
[1 0 0;
 0 1 1;
 0 1 0]

or
[0 1 0;
 1 1 0;
 0 0 1]

depending on whether the iteration index is odd or even. For example, the matrix product [1 3 2]*[0 1 0; 1 1 0; 0 0 1] gives [3 4 2]. Then [3,4,2]*[1 0 0; 0 1 1; 0 1 0] gives [3 6 4], and so on.
Note also that the second matrix equals the first rotated 180 degrees, which can be exploited to save a few bytes.
7        % Push 7
B        % Convert to binary. Gives [1 1 1]. This is the initial level
i        % Take input n
:        % Push range [1 2 ... n]
&quot;        % For each
  5      %   Push 5
  I:     %   Push range [1 2 3]
  -      %   Subtract, element-wise: gives [4 3 2]
  B      %   Convert to binary. This gives the matrix [1 0 0; 0 1 1; 0 1 0]
  @      %   Push current iteration index
  E      %   Multiply by 2. Gives 2 in the firt iteration, 4 in the second etc
  X!     %   Rotate matrix 90 degrees either 2 or 0 times
  Y*     %   Matrix multiply
         % End. Implicitly display