Golf a Numerical Growing Braid
MATL, 18 17 16 bytes
7Bi:&quot;2:4PB@EX!Y*

Input is 0-based.
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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