This is undoubtedly related to the fascnating suibject of this: https://arxiv.org/pdf/0806.3583.pdf Victor On Mon, Sep 3, 2018 at 1:22 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
I just noticed that the sum of the digits of each of the numbers 0 through 13 was the same in base -2 and in base 4. Why should that be? There's no obvious pattern linking them. And the rule is broken with 14.
0 0 0 0 0 1 1 1 1 1 2 110 2 2 2 3 111 3 3 3 4 100 1 10 1 5 101 2 11 2 6 11010 3 12 3 7 11011 4 13 4 8 11000 2 20 2 9 11001 3 21 3 10 11110 4 22 4 11 11111 5 23 5 12 11100 3 30 3 13 11101 4 31 4 14 10010 2 32 5
Those digit sums are A027615 and A053737. A276328 and A276332 also start with those same 13 terms.
Through 100, the base 4 sum of digits agrees with the base -2 sum of digits 81 times. Through 1000, they agree 560 times. Through 10000, they agree 4891 times. When they disagree, the latter always exceeds the former, and does so by a small multiple of 3 (at least through 10000, in which range the maximum difference is 15).
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