Writing a program to illustrate a fractal

[Musatov]

I have an integer sequence of a fractal nature and want to know if it is possible to write a program to illustrate it in a manner similar to the many animated Mandelbrot illustrations.

The sequence is defined by:

For 1 <= n <= 3, a(n) = n; thereafter, a(2n) = a(n) + a(n+1), a(2n-1) = a(n) + a(n-2).

Output begins:
1, 2, 3, 5, 4, 8, 7, 9, 7, 12, 13, 15, 11, 16, 17, 16, 14, 19, 21, 25, 20, 28, 27, 26, 24, 27, 31, 33...