Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is
1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . .
If you delete the first occurrence of each positive integer, you'll see that the remaining sequence is the same as the original. (So, if you do it again and again, you always get the same sequence.)

@online{FractalSequences,
title = {Fractal Sequences},
abstract = { Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is
1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . .
If you delete the first occurrence of each positive integer, you'll see that the remaining sequence is the same as the original. (So, if you do it again and again, you always get the same sequence.)},
url = {https://faculty.evansville.edu/ck6/integer/fractals.html},
year = 1999,
author = {Clark Kimberling},
comment = {},
urldate = {2019-08-21},
collections = {easily-explained,fun-maths-facts,integerology,puzzles}
}