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A troublesome sequence
  A number sequence is a set of numbers arranged in an orderly fashion, such that the preceding and following numbers are completely specified. Sometimes it is very easy to find in a series what number comes next, but usually it is not! Here is a tough example: try to replace the ‘X’ in the following sequence with the most appropriate number:
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, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, X, ...
Can you guess the secret rule and the magic of the sequence above?

Difficulty level: , general math knowledge.
Category: Number series.
Keywords: number sequence, progression, series.
Related puzzles:
- The Parrot sequence,
- Pacioli puzzle.

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The answer is X = 6.

Remove every alternating (second) number of this special sequence, what do you have left? 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8... Exactly the same sequence! Which means if you remove every second term again, you get the same sequence. Over and over.
This sequence has in fact the property to contain itself as a proper subsequence, infinitely. This is why it is called ‘fractal sequence’ or ‘sandwich sequence’.

Vivisection of the sequence
Curiously enough, there are actually infinite positive integer subsequences embedded in this fractal sequence (see table further below):
Subsequence a: Starts from position n = 1 and increments by 1 while moving 2 numbers ahead in the sequence.
Subsequence b: Starts from position n = 2 and increments by 1 while moving 2²=4 numbers ahead in the sequence.
Subsequence c: Starts from position n = 4 and increments by 1 while moving 2³=8 numbers ahead in the sequence.
Subsequence d: Starts from position n = 8 and increments by 1 while moving 2⁴=16 numbers ahead in the sequence.
Etc...
As you can see in the table below, each integer subsequence starts on the 2^m-1 position and jumps ahead in the fractal sequence by 2^m (with m≥0) positions and increments by 1.

The 5 Winners of the Puzzle of the Month are:
Larry Bickford, USA - Emeline Luirard, France - Denzil Gumbo, Zimbabwe - Jakub Nogly, Poland - Sarah Farooq, Pakistan

Congratulations!

Properties of the sequence
This particular fractal sequence is obtained from powers of 2. In fact, every number of the sequence occurs at 2^m(2k - 1) position, with m≥0.
For instance, the number 6 occurs at the following positions (n):
n₁ = 2⁰(2x6 - 1) = 11
n₂ = 2¹(2x6 - 1) = 22
n₃ = 2²(2x6 - 1) = 44
Etc...

Here is a simple program in "bc" (available on Unix, Linux, and Cygwin) sent by Larry Bickford that generates the sequence:

With i < 50, we obtain the following output:
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 16 1 17 9 18 5 19 10 20 3 21 11 22 6 23 12 24 2 25 13

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