19 NEW PRENUMBERED SEQUENCES A125991-A126005, A126010-A126013 FOLLOW.
I leave it up to you whether these should have the keyword "base",
as the encoding used (explained in
http://www.research.att.com/~njas/sequences/A106486 )
involves the binary digits of n.
Note however, the similar encoding for impartial games
described in http://www.research.att.com/~njas/sequences/A034797
That sequence and related http://www.research.att.com/~njas/sequences/A034798
and http://www.research.att.com/~njas/sequences/A079599
do not use the keyword "base".
%I A125991
%S A125991 0,8,16,24,64,72,80,88,128,136,144,152,192,200,208,216,512,520,528,
%T A125991 536,576,584,592,600,640,648,656,664,704,712,720,728,2048,2056,2064,
%U A125991 2072,2112,2120,2128,2136,2176,2184,2192,2200,2240,2248,2256,2264
%N A125991 A106486-encodings of combinatorial games with zero value.
%C A125991 In these games, the second player can always win.
%e A125991 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.
%H A125991 A. Karttunen, Scheme-program for computing this sequence.
%Y A125991 Row 1 of A126000. Intersection of A126001 and A126002. Characteristic function occurs as row 0 of A126010.
%K A125991 nonn
%O A125991 1,2
%A A125991 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125992
%S A125992 1,17,65,81,513,529,577,593,2049,2065,2113,2129,2561,2577,2625,2641,
%T A125992 4097,4113,4161,4177,4609,4625,4673,4689,6145,6161,6209,6225,6657,
%U A125992 6673,6721,6737,8193,8209,8257,8273,8705,8721,8769,8785,10241,10257
%N A125992 A106486-encodings of combinatorial games with value 1.
%C A125992 These are codes for games which belong to the same equivalence class as the game {0|} (i.e. game 1).
%e A125992 Game {0|} is encoded as 2^(2*0) = 1, thus 1 is the first term of this sequence. Also 17 belongs into this sequence, as it encodes game {-1,0|}, where, as the option -1 is dominated by option 0, the former can be deleted, resulting the same game {0|}. Also code 65536 (= 2^(2*(2^(1+2*1)))) belongs into this sequence, as it encodes the game {{|1}|}, which is reversible to game 1.
%H A125992 A. Karttunen, Scheme-program for computing this sequence.
%Y A125992 Row 2 of A126000.
%K A125992 nonn
%O A125992 1,2
%A A125992 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125993
%S A125993 2,10,130,138,514,522,642,650,2050,2058,2178,2186,2562,2570,2690,2698,
%T A125993 8194,8202,8322,8330,8706,8714,8834,8842,10242,10250,10370,10378,
%U A125993 10754,10762,10882,10890,32770,32778,32898,32906,33282,33290,33410
%N A125993 A106486-encodings of combinatorial games with value -1.
%C A125993 These are codes for games which belong to the same equivalence class as the game {|0} (i.e. game -1).
%e A125993 Game {|0} is encoded as 2^(1+2*0) = 2, thus 2 is the first term of this sequence. Also 10 belongs belongs into this sequence, as it encodes game {|0,1}, where, as the option 0 dominates the option 1, the latter can be deleted, resulting the same game {|0}. Likewise code 8589934592 (= 2^(1+(2*2^(2*2)))) belongs into this sequence, as it encodes the game {|{-1|}}, which is reversible to game -1.
%H A125993 A. Karttunen, Scheme-program for computing this sequence.
%Y A125993 Row 3 of A126000.
%K A125993 nonn
%O A125993 1,1
%A A125993 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125994
%S A125994 3,11,19,27,515,523,531,539,2051,2059,2067,2075,2563,2571,2579,2587,
%T A125994 8195,8203,8211,8219,8707,8715,8723,8731,10243,10251,10259,10267,
%U A125994 10755,10763,10771,10779,32771,32779,32787,32795,33283,33291,33299
%N A125994 A106486-encodings of combinatorial games equivalent to game {0|0}.
%C A125994 These are codes for games which belong to the same equivalence class as the game {0|0} (i.e. game *).
%e A125994 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}.
%H A125994 A. Karttunen, Scheme-program for computing this sequence.
%Y A125994 Row 4 of A126000. Subset of A126003.
%K A125994 nonn
%O A125994 1,1
%A A125994 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125995
%S A125995 4,5,20,21,68,69,84,85,4100,4101,4116,4117,4164,4165,4180,4181,16388,
%T A125995 16389,16404,16405,16452,16453,16468,16469,20484,20485,20500,20501,
%U A125995 20548,20549,20564,20565,65540,65541,65556,65557,65604,65605,65620
%N A125995 A106486-encodings of combinatorial games with value 2.
%C A125995 These are codes for games which belong to the same equivalence class as the game {1|} (i.e. the game 2).
%e A125995 Game {1|} is encoded as 2^(2*1) = 4, thus 4 is the first term of this sequence. Also 5 belongs into this sequence, as it encodes game {0,1|}, where, because the option 1 dominates the option 0 on the left side, the zero can be deleted, resulting the same game {1|}.
%H A125995 A. Karttunen, Scheme-program for computing this sequence.
%Y A125995 Row 5 of A126000.
%K A125995 nonn
%O A125995 1,1
%A A125995 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125996
%S A125996 6,7,14,15,22,23,30,31,70,71,78,79,86,87,94,95,518,519,526,527,534,
%T A125996 535,542,543,582,583,590,591,598,599,606,607,2054,2055,2062,2063,2070,
%U A125996 2071,2078,2079,2118,2119,2126,2127,2134,2135,2142,2143,2566,2567
%N A125996 A106486-encodings of combinatorial games equivalent to game {1|0}.
%C A125996 These are codes for games which belong to the same equivalence class as the game {1|0}.
%e A125996 Game {1|0} is encoded as 2^(2*1) + 2^(1+2*0) = 6, thus 6 is the first term of this sequence. Also 7 belongs into this sequence, as it encodes game {0,1|0}, where, as the option 1 dominates the option 0 on the left side, the former can be deleted, resulting the same game {1|0}.
%H A125996 A. Karttunen, Scheme-program for computing this sequence.
%Y A125996 Row 6 of A126000.
%K A125996 nonn
%O A125996 1,1
%A A125996 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125997
%S A125997 9,25,73,89,521,537,585,601,2057,2073,2121,2137,2569,2585,2633,2649,
%T A125997 4105,4121,4169,4185,4617,4633,4681,4697,6153,6169,6217,6233,6665,
%U A125997 6681,6729,6745,8201,8217,8265,8281,8713,8729,8777,8793,10249,10265
%N A125997 A106486-encodings of combinatorial games equivalent to game {0|1}.
%C A125997 These are codes for games which belong to the same equivalence class as the game {0|1} (game 1/2).
%e A125997 Game {0|1} is encoded as 2^(2*0) + 2^(1+2*1) = 9, thus 9 is the first term of this sequence. Also 25 (= 2^(2*2) + 2^(2*0) + 2^(1+2*1)) belongs into this sequence, as it encodes game {-1,0|1}, where, as the option -1 is dominated by option 0, the former can be deleted, resulting the same game {0|1}.
%H A125997 A. Karttunen, Scheme-program for computing this sequence.
%Y A125997 Row 7 of A126000.
%K A125997 nonn
%O A125997 1,1
%A A125997 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125998
%S A125998 12,13,28,29,76,77,92,93,524,525,540,541,588,589,604,605,2060,2061,
%T A125998 2076,2077,2124,2125,2140,2141,2572,2573,2588,2589,2636,2637,2652,
%U A125998 2653,4108,4109,4124,4125,4172,4173,4188,4189,4620,4621,4636,4637
%N A125998 A106486-encodings of combinatorial games equivalent to game {1|1}.
%C A125998 These are codes for games which belong to the same equivalence class as the game {1|1}, the impartial game 1*.
%e A125998 Game {1|1} is encoded as 2^(2*1) + 2^(1+2*1) = 12, thus 12 is the first term of this sequence. Also 13 belongs into this sequence, as it encodes game {0,1|1}, where, as the option 0 is dominated by option 1, the former can be deleted, resulting the same game {1|1}.
%H A125998 A. Karttunen, Scheme-program for computing this sequence.
%Y A125998 Row 8 of A126000.
%K A125998 nonn
%O A125998 1,1
%A A125998 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126001
%S A126001 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,
%T A126001 81,84,85,88,89,92,93,128,129,132,133,136,137,140,141,144,145,148,149,
%U A126001 152,153,156,157,192,193,196,197,200,201,204,205,208,209,212,213,216
%N A126001 A106486-encodings of nonnegative combinatorial games, i.e. games whose value is >= 0.
%C A126001 In these games, the left can always win if he is to play second.
%H A126001 A. Karttunen, Scheme-program for computing this sequence.
%Y A126001 Characteristic function occurs as row 0 of A125999. Cf. A125991, A126003-A126005.
%K A126001 nonn
%O A126001 1,3
%A A126001 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126002
%S A126002 0,2,8,10,16,18,24,26,32,34,40,42,48,50,56,58,64,66,72,74,80,82,88,90,
%T A126002 96,98,104,106,112,114,120,122,128,130,136,138,144,146,152,154,160,
%U A126002 162,168,170,176,178,184,186,192,194,200,202,208,210,216,218,224,226
%N A126002 A106486-encodings of combinatorial games whose value is <= 0.
%C A126002 In these games, the right can always win if he is to play second.
%H A126002 A. Karttunen, Scheme-program for computing this sequence.
%Y A126002 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.
%K A126002 nonn
%O A126002 1,2
%A A126002 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126003
%S A126003 3,6,7,11,14,15,19,22,23,27,30,31,33,35,36,37,38,39,41,43,44,45,46,47,
%T A126003 49,51,52,53,54,55,57,59,60,61,62,63,67,70,71,75,78,79,83,86,87,91,94,
%U A126003 95,97,99,100,101,102,103,105,107,108,109,110,111,113,115,116,117,118
%N A126003 A106486-encodings of combinatorial games whose value is incomparable with zero game, i.e. fuzzy games.
%C A126003 In these games, the first player can always win.
%H A126003 A. Karttunen, Scheme-program for computing this sequence.
%Y A126003 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.
%K A126003 nonn
%O A126003 1,1
%A A126003 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126004
%S A126004 1,4,5,9,12,13,17,20,21,25,28,29,65,68,69,73,76,77,81,84,85,89,92,93,
%T A126004 129,132,133,137,140,141,145,148,149,153,156,157,193,196,197,201,204,
%U A126004 205,209,212,213,217,220,221,256,257,260,261,264,265,268,269,272,273
%N A126004 A106486-encodings of combinatorial games whose value is greater than zero.
%C A126004 In these games, the left can always win.
%H A126004 A. Karttunen, Scheme-program for computing this sequence.
%Y A126004 Intersection of complement of A126002 and A126001. Cf. A125991, A126001-A126003.
%K A126004 nonn
%O A126004 1,2
%A A126004 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126005
%S A126005 2,10,18,26,32,34,40,42,48,50,56,58,66,74,82,90,96,98,104,106,112,114,
%T A126005 120,122,130,138,146,154,160,162,168,170,176,178,184,186,194,202,210,
%U A126005 218,224,226,232,234,240,242,248,250,514,522,530,538,544,546,552,554
%N A126005 A106486-encodings of combinatorial games whose value is less than zero.
%C A126005 In these games, the right can always win.
%H A126005 A. Karttunen, Scheme-program for computing this sequence.
%Y A126005 Intersection of complement of A126001 and A126002. Cf. A125991, A126001-A126003.
%K A126005 nonn
%O A126005 1,1
%A A126005 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126010
%S A126010 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,
%T A126010 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,
%U A126010 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
%N 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), ...
%C A126010 Here we use the encoding described in A106486.
%e A126010 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.
%H A126010 A. Karttunen, Scheme-program for computing this sequence.
%Y A126010 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.
%K A126010 nonn,tabl
%O A126010 0,1
%A A126010 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A125999
%S A125999 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,
%T A125999 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,
%U A125999 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
%N 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), ...
%C A125999 Here we use the encoding explained in A106486. A(i,j) = A(A106485(j),A106485(i)).
%H A125999 A. Karttunen, Scheme-program for computing this sequence.
%Y A125999 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.
%K A125999 nonn,tabl
%O A125999 0,1
%A A125999 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126011
%S A126011 0,1,2,3,4,6,9,12,18,32,33,36,48,66,67,96,97,129,131,132,134,195,256,
%T A126011 258,264,288,384,386,516,768,4098,4099,4102,4128,4129,4132,4227,4230,
%U A126011 8196,8198,8204,8448,8450,8456,12294
%N A126011 A106486-encodings for the minimal representatives of each equivalence class of the finite combinatorial games.
%C A126011 The initial terms correspond with the following games: code 0 = {|} = 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.
%H A126011 D. Eppstein, Combinatorial Game Theory links
%H A126011 A. Karttunen, Scheme-program for computing this sequence.
%H A126011 A. Siegel, Combinatorial Game Suite
%H A126011 D. Wolfe, Several publications about combinatorial game theory
%H A126011 Wikipedia-article about combinatorial game theory
%D A126011 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol 1, A K Peters, 2001.
%D A126011 John H. Conway, On Numbers and Games, Second Edition, A K Peters, 2001.
%Y A126011 Records of A126012. Column 1 of A126000. Inverse: A126013. See also A126010 and A065401.
%Y A126011 Sequences A034797, A034798, A079599 utilize a similar encoding system for impartial games.
%K A126011 nonn
%O A126011 0,3
%A A126011 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126012
%S A126012 0,1,2,3,4,4,6,6,0,9,2,3,12,12,6,6,0,1,18,3,4,4,6,6,0,9,18,3,12,12,6,
%T A126012 6,32,33,32,33,36,36,36,36,32,33,32,33,36,36,36,36,48,33,48,33,36,36,
%U A126012 36,36,48,33,48,33,36,36,36,36,0,1,66,67,4,4,6,6,0,9,66,67,12,12,6,6
%N A126012 A106486-encoding of the canonical representative of the combinatorial game with code n.
%e A126012 25 (= 2^(2*2) + 2^(2*0) + 2^(1+2*1)) encodes the game {-1,0|1}, where, as the option -1 is dominated by option 0, the former can be deleted, giving us the game {0|1}, i.e. the canonical (minimal) form of the game 1/2, encoded as 2^(2*0) + 2^(1+2*1) = 9, thus a(25)=9 and a(9)=9. Similarly a(65536)=1, as 65536 (= 2^(2*(2^(1+2*1)))) encodes the game {{|1}|}, which is reversible to the game {0|}, i.e. the game 1, which is encoded as 2^(2*0) = 1.
%H A126012 A. Karttunen, Scheme-program for computing this sequence.
%Y A126012 A126011 gives the distinct terms (and also the records).
%K A126012 nonn
%O A126012 0,3
%A A126012 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126013
%S A126013 0,1,2,3,4,0,5,0,0,6,0,0,7,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,9,10,
%T A126013 0,0,11,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,
%U A126013 14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,16,0,0
%N A126013 Inverse function of N->N injection A126011.
%C A126013 a(0)=0 because A126011(0)=0, but a(n) = 0 also for those n which do not occur as values of A126011. All positive natural numbers occur here once.
%H A126013 A. Karttunen, Scheme-program for computing this sequence.
%Y A126013 a(A126011(n)) = n for all n.
%K A126013 nonn
%O A126013 0,3
%A A126013 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006
%I A126000
%S A126000 0,8,1,16,17,2,24,65,10,3,64,81,130,11,4,72,513,138,19,5,6,80,529,514,
%T A126000 27,20,7,9,88,577,522,515,21,14,25,12,128,593,642,523,68,15,73,13,18,
%U A126000 136,2049,650,531,69,22,89,28,26,32,144,2065,2050,539,84,23,521,29
%N A126000 Table giving A106486-codes for each equivalence class of combinatorial games.
%C A126000 See the comments and references on A126011 and on the individual rows.
%e A126000 Each row lists the integers that encode the games with the same value as the initial term of the row:
%e A126000 0,8,16,24,64,72,80,88,128,136,144,152,192,200,208,...
%e A126000 1,17,65,81,513,529,577,593,2049,2065,2113,2129,2561,...
%e A126000 2,10,130,138,514,522,642,650,2050,2058,2178,2186,...
%e A126000 3,11,19,27,515,523,531,539,2051,2059,2067,2075,2563,...
%e A126000 4,5,20,21,68,69,84,85,4100,4101,4116,4117,4164,4165,...
%e A126000 6,7,14,15,22,23,30,31,70,71,78,79,86,87,94,95,518,...
%e A126000 9,25,73,89,521,537,585,601,2057,2073,2121,2137,2569,...
%e A126000 12,13,28,29,76,77,92,93,524,525,540,541,588,589,604,...
%e A126000 18,26,146,154,530,538,658,666,2066,2074,2194,2202,...
%e A126000 32,34,40,42,160,162,168,170,544,546,552,554,672,674,...
%e A126000 33,35,41,43,49,51,57,59,161,163,169,171,177,179,185,...
%e A126000 36,37,38,39,44,45,46,47,52,53,54,55,60,61,62,63,100,...
%e A126000 48,50,56,58,176,178,184,186,560,562,568,570,688,690,...
%e A126000 66,74,82,90,194,202,210,218,578,586,594,602,706,714,...
%H A126000 A. Karttunen, Scheme-program for computing this sequence.
%H A126000 Index entries for sequences that are permutations of the natural numbers
%Y A126000 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.
%K A126000 nonn,tabl
%O A126000 1,2
%A A126000 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Dec 18 2006