A125991 -id:A125991 - cp's OEIS frontend

A126000 Table giving A106486-codes for each equivalence class of combinatorial games.

0, 8, 1, 16, 17, 2, 24, 65, 10, 3, 64, 81, 130, 11, 4, 72, 513, 138, 19, 5, 6, 80, 529, 514, 27, 20, 7, 9, 88, 577, 522, 515, 21, 14, 25, 12, 128, 593, 642, 523, 68, 15, 73, 13, 18, 136, 2049, 650, 531, 69, 22, 89, 28, 26, 32, 144, 2065, 2050, 539, 84, 23, 521, 29

Offset: 1

Antti Karttunen, Dec 18 2006

See the comments and references on A126011 and on the individual rows.

			Each row lists the integers that encode the games with the same value as the initial term of the row:
0,8,16,24,64,72,80,88,128,136,144,152,192,200,208,...
1,17,65,81,513,529,577,593,2049,2065,2113,2129,2561,...
2,10,130,138,514,522,642,650,2050,2058,2178,2186,...
3,11,19,27,515,523,531,539,2051,2059,2067,2075,2563,...
4,5,20,21,68,69,84,85,4100,4101,4116,4117,4164,4165,...
6,7,14,15,22,23,30,31,70,71,78,79,86,87,94,95,518,...
9,25,73,89,521,537,585,601,2057,2073,2121,2137,2569,...
12,13,28,29,76,77,92,93,524,525,540,541,588,589,604,...
18,26,146,154,530,538,658,666,2066,2074,2194,2202,...
32,34,40,42,160,162,168,170,544,546,552,554,672,674,...
33,35,41,43,49,51,57,59,161,163,169,171,177,179,185,...
36,37,38,39,44,45,46,47,52,53,54,55,60,61,62,63,100,...
48,50,56,58,176,178,184,186,560,562,568,570,688,690,...
66,74,82,90,194,202,210,218,578,586,594,602,706,714,...

Column 1: A126011. Row 1: A125991, row 2: A125992, row 3: A125993, row 4: A125994, row 5: A125995, row 6: A125996, row 7: A125997, row 8: A125998.

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

Original entry on oeis.org

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++);

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.

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.

A126004 A106486-encodings of combinatorial games whose value is greater than zero.

Original entry on oeis.org

1, 4, 5, 9, 12, 13, 17, 20, 21, 25, 28, 29, 65, 68, 69, 73, 76, 77, 81, 84, 85, 89, 92, 93, 129, 132, 133, 137, 140, 141, 145, 148, 149, 153, 156, 157, 193, 196, 197, 201, 204, 205, 209, 212, 213, 217, 220, 221, 256, 257, 260, 261, 264, 265, 268, 269, 272, 273

Offset: 1

Antti Karttunen, Dec 18 2006

nonn

In these games, the left can always win.

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

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

cp's OEIS Frontend

A126000 Table giving A106486-codes for each equivalence class of combinatorial games.

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

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

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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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A126004 A106486-encodings of combinatorial games whose value is greater than zero.

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