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

Author

Antti Karttunen, Dec 18 2006

Keywords

• nonn

Comments

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.

References

• 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.

Links

• 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

Crossrefs

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.

Extensions

Table of terms added Jan 01 2007.