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A018219 Table T(a,b) by antidiagonals of winning positions in 3-pile Wythoff game (a square array). 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 21 13:24 EDT 2019. Contains 328299 sequences. (Running on oeis4.)