[seqfan] Sum of digits in base -2 and base 4
Keith F. Lynch
kfl at KeithLynch.net
Mon Sep 3 15:39:31 CEST 2018
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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