The other day on a train I was investigating the different ways that numbers can be represented using an “extended binary” when I happened upon an interesting sequence.

Extended binary has the same values for the digits (2^{0},
2^{1},
2^{2}, 2^{3}…), but instead of allowing only
coefficients of 0 and 1, we allow any natural number. So the number
6 can be represented in any of the following ways:

- 1·2
^{2}+ 1·2^{1} - 1·2
^{2}+ 2·2^{0} - 3·2
^{1} - 2·2
^{1}+ 2·2^{0} - 1·2
^{1}+ 4·2^{0}

So the number six has five representations. If we count the number of representations for each number, starting with 0, we get the following sequence:

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 13, 18, 18, 23, 23, 30, 30, 37, 37, 47, 47, 57, 57, …

The first thing I do when analyzing a sequence is to look at the differences between the successive numbers. In this case every number repeats, but if we remove the repeats, like so:

1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, …

And then take the difference between each of these numbers, we get:

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, …

Sloane’s has it as A040039, with the subtitle:

First differences of A033485; also A033485 with terms repeated.