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%I #6 Mar 30 2012 17:37:52
%S 0,0,0,1,0,0,1,0,2,3,0,3,2,1,0,0,1,0,2,3,0,3,2,1,0,4,5,6,7,0,5,4,7,6,
%T 1,0,6,7,4,5,2,3,0,7,6,5,4,3,2,1,0,0,1,0,2,3,0,3,2,1,0,4,5,6,7,0,5,4,
%U 7,6,1,0,6,7,4,5,2,3,0,7,6,5,4,3,2,1,0,8,9,10,11,12,13,14,15,0,9,8,11,10,13
%N Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives x values.
%C (x,y,z) is a losing position iff the mod-2 sum of the binary expansions of x,y,z (without carries) is 0.
%C In this sort the first few triples are: 0 0 0, 0 1 1, 0 2 2, 1 2 3, 0 3 3, 0 4 4, 1 4 5, 0 5 5, 2 4 6, 3 5 6, 0 6 6, 3 4 7, 2 5 7, 1 6 7, 0 7 7, 0 8 8, 1 8 9, 0 9 9, 2 8 10, 3 9 10, 0 10 10, 3 8 11, 2 9 11, 1 10 11, 0 11 11, 4 8 12, 5 9 12, 6 10 12, 7 11 12, 0 12 12. The 0,0,0 triple was added by Joshua Zucker.
%D I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.
%Y Cf. A080594, A080595.
%Y A119464, A119465, A119466 give the same terms as these sequences but sorted in a different order (by sum rather than by value of z).
%K easy,nonn
%O 0,9
%A _N. J. A. Sloane_, Feb 23 2003
%E Corrected and extended by _John W. Layman_, Oct 22 2003
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