A126000 -id:A126000 - cp's OEIS frontend

A126001 A106486-encodings of nonnegative combinatorial games, i.e., games whose value is >= 0.

0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 128, 129, 132, 133, 136, 137, 140, 141, 144, 145, 148, 149, 152, 153, 156, 157, 192, 193, 196, 197, 200, 201, 204, 205, 208, 209, 212, 213, 216

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the left can always win if he is to play second.

Antti Karttunen, Table of n, a(n) for n = 1..16386
A. Karttunen, Scheme-program for computing this sequence.

Characteristic function (A320006) occurs as row 0 of A125999.

Cf. A125991, A126003, A126004, A126005.

A320006(n) = if(!n,1,my(m=(n>>1), r=0); while(m>0, if((m%2)&&!A320007(r),return(0)); m >>= 2; r++); (1));
A320007(n) = if(!n,0,my(m=n, s=0); while(m>0, if((m%2)&&A320006(s),return(1)); m >>= 2; s++); (0));
k=0; n=0; while(k<16386, if(A320006(n), k++; write("b126001.txt", k, " ", n)); n++);

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

Original entry on oeis.org

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.

A126002 A106486-encodings of combinatorial games whose value is <= 0.

Original entry on oeis.org

0, 2, 8, 10, 16, 18, 24, 26, 32, 34, 40, 42, 48, 50, 56, 58, 64, 66, 72, 74, 80, 82, 88, 90, 96, 98, 104, 106, 112, 114, 120, 122, 128, 130, 136, 138, 144, 146, 152, 154, 160, 162, 168, 170, 176, 178, 184, 186, 192, 194, 200, 202, 208, 210, 216, 218, 224, 226, 232, 234, 240, 242, 248, 250, 512, 514

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the right can always win if he is to play second.

Antti Karttunen, Table of n, a(n) for n = 1..16384
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences which agree for a long time but are different

Characteristic function occurs as column 0 of A125999. Differs from A047467 (a(65)=512, not 256) and also from A079599, as the term 18446744073709551616 (= 2^64) is a member of this sequence but not of A079599. Cf. A125991, A126003-A126005.

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.

A126003 A106486-encodings of combinatorial games whose value is incomparable with zero game, i.e., fuzzy games.

Original entry on oeis.org

3, 6, 7, 11, 14, 15, 19, 22, 23, 27, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 67, 70, 71, 75, 78, 79, 83, 86, 87, 91, 94, 95, 97, 99, 100, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 116, 117, 118

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the first player can always win.

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

Intersection of the complements of A126001 and A126002, or equally, complement of the union of A126001 and A126002. Differs from A047556. Cf. A125991, A125994, A126004-A126005.

A126010 Square array A(g,h) = 1 if combinatorial games g and h have the same value, 0 if they differ, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Antti Karttunen, Dec 18 2006

Here we use the encoding described in A106486.

			A(4,5) = A(5,4) = 1 because 5 encodes the game {0,1|}, where, because the option 1 dominates the option 0 on the left side, the zero can be deleted, resulting the game {1|}, the canonical form of the game 2, which is encoded as 4.

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

Row 0 is the characteristic function of A125991 (shifted one step). A(i, j) = A125999(i, j)*A125999(j, i). A126011 gives the A106486-encodings for the minimal representatives of each equivalence class of finite combinatorial games.

A126005 A106486-encodings of combinatorial games whose value is less than zero.

Original entry on oeis.org

2, 10, 18, 26, 32, 34, 40, 42, 48, 50, 56, 58, 66, 74, 82, 90, 96, 98, 104, 106, 112, 114, 120, 122, 130, 138, 146, 154, 160, 162, 168, 170, 176, 178, 184, 186, 194, 202, 210, 218, 224, 226, 232, 234, 240, 242, 248, 250, 514, 522, 530, 538, 544, 546, 552, 554

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the right can always win.

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

Intersection of complement of A126001 and A126002. Cf. A125991, A126001-A126003.

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).

A125994 A106486-encodings of combinatorial games equivalent to game {0|0}.

Original entry on oeis.org

3, 11, 19, 27, 515, 523, 531, 539, 2051, 2059, 2067, 2075, 2563, 2571, 2579, 2587, 8195, 8203, 8211, 8219, 8707, 8715, 8723, 8731, 10243, 10251, 10259, 10267, 10755, 10763, 10771, 10779, 32771, 32779, 32787, 32795, 33283, 33291, 33299

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

These are codes for games which belong to the same equivalence class as the game {0|0} (i.e. game *).

			Game {0|0} is encoded as 2^(2*0) + 2^(1+2*0) = 3, thus 3 is the first term of this sequence. However, 11 also belongs into this sequence, as it encodes game {0|0,1}, where, because the option 0 dominates the option 1 on the right hand side, the latter can be deleted, resulting the same game {0|0}.

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

Row 4 of A126000. Subset of A126003.

cp's OEIS Frontend

A126001 A106486-encodings of nonnegative combinatorial games, i.e., games whose value is >= 0.

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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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A126002 A106486-encodings of combinatorial games whose value is <= 0.

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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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A126003 A106486-encodings of combinatorial games whose value is incomparable with zero game, i.e., fuzzy games.

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A126010 Square array A(g,h) = 1 if combinatorial games g and h have the same value, 0 if they differ, 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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A126005 A106486-encodings of combinatorial games whose value is less than zero.

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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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A125994 A106486-encodings of combinatorial games equivalent to game {0|0}.

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