A080593 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.

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, 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, 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

Offset

0,9

Comments

(x,y,z) is a losing position iff the mod-2 sum of the binary expansions of x,y,z (without carries) is 0.

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.

References

I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

Links

Table of n, a(n) for n=0..100.

Crossrefs

Cf. A080594, A080595.

A119464, A119465, A119466 give the same terms as these sequences but sorted in a different order (by sum rather than by value of z).

Sequence in context: A051910 A137998 A316590 * A319148 A193682 A051933

Adjacent sequences: A080590 A080591 A080592 * A080594 A080595 A080596

Keyword

easy,nonn

Author

N. J. A. Sloane, Feb 23 2003

Extensions

Corrected and extended by John W. Layman, Oct 22 2003

Status

approved