A018219 Table T(a,b) by antidiagonals of winning positions in 3-pile Wythoff game (a square array).

0, 2, 2, 1, 4, 1, 5, 0, 0, 5, 7, 3, 6, 3, 7, 3, 1, 8, 8, 1, 3, 10, 6, 10, 1, 10, 6, 10, 4, 5, 12, 4, 4, 12, 5, 4, 13, 12, 2, 0, 3, 0, 2, 12, 13, 15, 15, 7, 9, 11, 11, 9, 7, 15, 15, 6, 17, 3, 11, 15, 7, 15, 11, 3, 17, 6, 18, 14, 11, 2, 0, 1, 1, 0, 2, 11, 14, 18, 20, 20, 4, 6, 19, 5, 11, 5, 19, 6

Offset: 0

Author

Michael Kleber

Keywords

• nonn
• tabl

Comments

(a,b,T(a,b)) are the winning positions in 3-pile Wythoff game. A move in k-pile Wythoff is: pick a subset of the k piles and remove the same number of stones from each. Goal: take the last stone.

T(a,b) = T(b,a). If T(a,b)=c then T(a,c)=b and T(b,c)=a.

Examples

			0 2 1 5 7 ...
2 4 0 3 1 ...
1 0 6 8 10 ...
5 3 8 1 4 ...
7 1 10 4 3 ...
T(1,1)=4, since from (114) your opponent can move to (113),(112),(111),(110),(014),(013),(004),(003). You can either win or move to (012) and win a move later.
		

Crossrefs

Rows 0-3: A002251, A018220-A018222. Main diagonal: A051261.

T(a, b)=0 iff A004481(a, b)=0 iff A002251(a)=b.

Programs

Mathematica

mex[ s_ ] := Min[ Complement[ Range[ 0, Max[ {s, -1} ]+1 ], Flatten[ s ] ] ]; f[ s_ ] := Join[ s, s+Table[ i, {i, Length[ s ]} ] ]; T[ a_, b_ ] := T[ a, b ] = mex[ { f[ Table[ T[ a-i, b ], {i, a} ] ], f[ Table[ T[ a, b-i ], {i, b} ] ], f[ Table[ T[ a-i, b-i ], {i, Min[ a, b ]} ] ] } ]

Extensions

Edited and extended by Christian G. Bower, Oct 29 2002