A126010 -id:A126010 - cp's OEIS frontend

A126011 A106486-encodings for the minimal representatives of each equivalence class of the finite combinatorial games.

0, 1, 2, 3, 4, 6, 9, 12, 18, 32, 33, 36, 48, 66, 67, 96, 97, 129, 131, 132, 134, 195, 256, 258, 264, 288, 384, 386, 516, 768, 4098, 4099, 4102, 4128, 4129, 4132, 4227, 4230, 8196, 8198, 8204, 8448, 8450, 8456, 12294, 262146, 262152, 262176, 262272

Offset: 0

Antti Karttunen, Dec 18 2006

nonn

The initial terms correspond with the following games: code 0 = {|} = the zero game, code 1 = {0|} = game 1, code 2 = {|0} = game -1, code 3 = {0|0} = game *, code 4 = {1|} = game 2, code 6 = {1|0}, code 9 = {0|1} = game 1/2, code 12 = {1|1} = game 1*, code 18 = {-1|0} = game -1/2, code 32 = {|-1} = game -2, code 33 = {0|-1}, code 36 = {1|-1} = game +-1, code 48 = {-1|-1} = game -1*, code 66 = {*|0} = game down, code 67 = {0,*|0} = game up*, code 96 = {*|-1}, code 97 = {0,*|-1}, code 129 = {0|*} = game up, code 131 = {0|0,*} = game down*, code 132 = {1|*}, code 134 = {1|0,*}, code 195 = {0,*|0,*} = game *2, code 256 = {2|} = game 3. Encoding is explained in A106486.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol. 1, A K Peters, 2001.
John H. Conway, On Numbers and Games, Second Edition, A K Peters, 2001.

A. Karttunen, Table of n, a(n) for n = 0..73
D. Eppstein, Combinatorial Game Theory links
A. Karttunen, Scheme-program for computing this sequence.
A. Siegel, Combinatorial Game Suite
Wikipedia, Combinatorial game theory
D. Wolfe, Several publications about combinatorial game theory

Records in A126012. Column 1 of A126000. Inverse: A126013. See also A126009 & A126010. A125990 gives the number of terms in range [0, 2^n[.

Sequences A034797, A034798, A079599 utilize a similar encoding system for impartial games.

Table of terms added Jan 01 2007.

A125991 A106486-encodings of combinatorial games with zero value.

Original entry on oeis.org

0, 8, 16, 24, 64, 72, 80, 88, 128, 136, 144, 152, 192, 200, 208, 216, 512, 520, 528, 536, 576, 584, 592, 600, 640, 648, 656, 664, 704, 712, 720, 728, 2048, 2056, 2064, 2072, 2112, 2120, 2128, 2136, 2176, 2184, 2192, 2200, 2240, 2248, 2256, 2264

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the second player can always win.

			Game 0 is encoded as zero, giving the first term of this sequence. Also 24 belongs into this sequence, as it encodes game {-1|1}, which the second player always wins. Similarly for game {*|*} which has code 2^(1+2*3) + 2^(2*3) = 192, thus 192 is a member of this sequence.

A. Karttunen, Scheme-program for computing this sequence.

Row 1 of A126000. Intersection of A126001 and A126002. Characteristic function occurs as row 0 of A126010.

A125999 Square array A(g,h) = 1 if combinatorial game g has value greater than or equal to that of game h, otherwise 0, listed antidiagonally in order A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Antti Karttunen, Dec 18 2006

Here we use the encoding explained in A106486. A(i,j) = A(A106485(j),A106485(i)).

A. Karttunen, Scheme-program for computing this sequence.

Row 0 is the characteristic function of A126001 (shifted one step) and similarly, column 0 is the characteristic function of A126002. Cf. tables A126010 and A126000.

A125990 Number of partisan games for which A106486-encoding of the minimal representation is less than 2^n.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 13, 17, 22, 28, 30, 30, 30, 38, 45, 45, 45, 45, 45, 53, 59, 59, 59, 59, 59

Offset: 0

Antti Karttunen, Jan 02 2007

Number of terms of A126011 in range [0,2^n[.

A. Karttunen, Scheme-program for computing this sequence.

Cf. A065401, A125999, A126010, A126011, A126012, A320006.

For all n, a(2*A114561(n)) = A065401(n).

cp's OEIS Frontend

A126011 A106486-encodings for the minimal representatives of each equivalence class of the finite combinatorial games.

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A125991 A106486-encodings of combinatorial games with zero value.

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A125999 Square array A(g,h) = 1 if combinatorial game g has value greater than or equal to that of game h, otherwise 0, listed antidiagonally in order A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

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A125990 Number of partisan games for which A106486-encoding of the minimal representation is less than 2^n.

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