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

0,2

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.

Links

Table of n, a(n) for n=0..87.

Example

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.

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 ]} ] ] } ]

Crossrefs

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

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

Sequence in context: A238606 A325612 A054995 * A174714 A116633 A263232

Adjacent sequences: A018216 A018217 A018218 * A018220 A018221 A018222

Keyword

nonn,tabl

Author

Michael Kleber

Extensions

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

Status

approved