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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 (2⁰, 2¹, 2², 2³…), 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:

• 1·2² + 1·2¹
• 1·2² + 2·2⁰
• 3·2¹
• 2·2¹ + 2·2⁰
• 1·2¹ + 4·2⁰

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.