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Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z sorted by sum, ties broken by putting smallest value of x first, then y, then z; sequence gives x values.
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%I #4 Mar 30 2012 17:36:54

%S 0,0,0,0,1,0,0,1,0,2,0,1,2,3,3,0,0,1,0,2,0,1,2,3,3,0,4,0,1,4,5,5,0,2,

%T 4,6,6,0,1,2,3,3,4,5,5,6,6,7,7,7,7,0,0,1,0,2,0,1,2,3,3,0,4,0,1,4,5,5,

%U 0,2,4,6,6,0,1,2,3,3,4,5,5,6,6,7,7,7,7,0,8,0,1,8,9,9,0,2,8,10,10,0,1,2,3,3

%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 sorted by sum, ties broken by putting smallest value of x first, then y, then z; sequence gives x values.

%C Thanks to _Ray Chandler_ for help in clarifying the relation between this sorted list and the one in A080593.

%e The triples with sum <= 20 (this sequence is the first column) are:

%e 0 0 0

%e 0 1 1

%e 0 2 2

%e 0 3 3

%e 1 2 3

%e 0 4 4

%e 0 5 5

%e 1 4 5

%e 0 6 6

%e 2 4 6

%e 0 7 7

%e 1 6 7

%e 2 5 7

%e 3 4 7

%e 3 5 6

%e 0 8 8

%e 0 9 9

%e 1 8 9

%e 0 10 10

%Y Cf. A119465, A119466 give the y and z values; A080593, A080594, A080595 give the same values sorted in a different way.

%K easy,nonn

%O 1,10

%A _Joshua Zucker_, May 21 2006