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%I #18 May 01 2016 09:02:01
%S 0,0,1,0,0,0,1,0,3,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,12,0,1,0,0,0,1,0,2,0,
%T 1,0,0,0,1,0,3,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,4,0,1,0,0,0,1,0,0,0,1,0,
%U 0,0,1,0,2,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,3,0,1,0,0,0,1,0,3,0,1,0,0,0,1,0,2,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,2,0,1,0,0,0,1,0,2
%N First row (row 0) of the Sprague-Grundy values of "3-pile Sharing Nim".
%C a(n) is G(0,0,n) = G(0,n,n) for 3-pile Sharing Nim.
%C If n is odd, a(n)=0. If n={4,12,16,20,28} (mod 32), a(n) is also 0.
%C If n=2 (mod 4), a(n)=1.
%C Thus all the "interesting" values are for n={0,8,24} (mod 32).
%C Ho conjectures that the sequence may be bounded. The highest value in the first thousand entries is a(24)=12.
%C Despite the simplicity of the "uninteresting" values, the "interesting" ones are provably not periodic, which means the entire sequence is not periodic.
%H Nhan Bao Ho, <a href="https://arxiv.org/abs/1506.06961">3-pile Sharing Nim and the linear time winning strategy</a>, arXiv:1506.06961 [math.CO], 2015, submitted to Theoretical Computer Science D. (Note that Ho indexes everything from 1, but the equations are simpler when indexed from 0, so I have used 0 in this entry.)
%F a(n)=0 if and only if n=0 or n=2^{2i}(2j+1) for some i,j>=0. This by itself proves the aperiodicity, since even the locations of the 0's are not periodic. Note that i=0 covers the "n is odd" case, i=1 covers the "n={4,12,20,28} (mod 32)" cases, i=2 covers the "n=16 (mod 32)" case, and i>=3 all fall under "n=0 (mod 32)". Thus the values for n={8,24} (mod 32) can never be 0.
%F The formula also implies that the limit of the density of zeros as n goes to infinity is 1/2 + 1/8 + 1/32 + 1/128 + ... = 2/3. - _Howard A. Landman_, Apr 20 2016
%e a(0)=a(1)=0 because there are no moves from (0,0,0) or (0,0,1). a(2)=1 because there is a move from (0,0,2) to (0,1,1), which has no moves and hence is value 0.
%K nonn,easy
%O 0,9
%A _Howard A. Landman_, Apr 10 2016
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