A126002 -id:A126002 - cp's OEIS frontend

A125991 A106486-encodings of combinatorial games with zero value.

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.

A079599 Numbers n for which the n-th impartial game is a second player win.

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: 0

Rob Arthan, Jan 28 2003

nonn

These are the indices n for which A034798(n) = 0.
From Antti Karttunen, Jan 30 2014: (Start)
A236678(a(n)) = n+1 for all n.
Differs from A047467 for the first time at a(64).
Differs from A126002(n+1) for the first time not later than at n=281474976710656 (= 2^48), as:
a((2^48)-1) = a(281474976710655) = 18085043209519168250 < 18446744073709551616 (= 2^64), while
a(2^48) = a(281474976710656) = 36893488147419103232 > 2^64.
(End)

			a(1) = 2 (rather than 1) because 1 = 2^0 = 2^a(0); a(64) = 512 (rather than 256) because 256 = 2^8 = 2^a(2).

J. H. Conway, On numbers and games.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences which agree for a long time but are different

Characteristic function: A236677, its partial sums: A236678.

Cf. A034798, A034797, A047467, A126002.

(define (A079599 n) (let loop ((n n) (i 0) (j 0) (s 0)) (cond ((zero? n) s) ((odd? n) (loop (/ (- n 1) 2) (+ i 1) (+ j 1 (A236677 j)) (+ s (expt 2 (+ j (A236677 j)))))) (else (loop (/ n 2) (+ i 1) (+ j 1 (A236677 j)) s)))))

a(0) = 0; a(n+1) = least x > a(n) such that the coefficient of 2^a(j) is zero in the binary expansion of x for all j < n+1

Alternatively: rewrite the binary representation of n in such a way that the forbidden bit-positions given by this sequence (with bit-position 0 standing for the least significant bit) are vacated, by shifting the rest of bits one bit left. E.g., bit-positions 0, 2, 8, 10, ... are forbidden, thus we rewrite 1 to 1x = 10 = 2, 2 (in binary 10) to 1x0x = 1000 = 8, 3 (in binary 11) to 1x1x = 1010 = 10, 4 (in binary 100) to 10x0x = 1000 = 16, 64 (in binary 1000000) to 1x00000x0x = 1000000000 = 512, etc. - Antti Karttunen, Jan 30 2014

More terms from Antti Karttunen, Jan 29 2014

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.

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

A125991 A106486-encodings of combinatorial games with zero value.

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A079599 Numbers n for which the n-th impartial game is a second player win.

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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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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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