cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A126013 Inverse function of N->N injection A126011.

Original entry on oeis.org

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, 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, 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
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2006

Keywords

Comments

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.

Crossrefs

a(A126011(n)) = n for all n.

A126009 Self-inverse permutation of integers induced when A106485 is restricted to A126011.

Original entry on oeis.org

0, 2, 1, 3, 9, 10, 8, 12, 6, 4, 5, 11, 7, 17, 18, 19, 20, 13, 14, 15, 16, 21
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

The Scheme-program given cannot in practice compute this further than n=21, as A106485(A126011(22))=36893488147419103232. However, the further terms could be deduced by other means. This sequence is a permutation of the nonnegative integers because combinatorial games form a group under (game) addition and each game has a well-defined, unique negative.

Formula

a(n) = A126013(A106485(A126011(n))).

Extensions

Typos corrected by Franklin T. Adams-Watters, Apr 06 2009

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 18 2006

Keywords

Comments

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

Examples

			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,...
		

Crossrefs

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.

A065401 Number of normal play partisan games born on or before day n.

Original entry on oeis.org

1, 4, 22, 1474
Offset: 0

Views

Author

R. K. Guy, Nov 23 2001

Keywords

Comments

Fraser and Wolfe prove upper and lower bounds on a(n+1) in terms of a(n). In particular they give the (probably quite weak) lower bound of 3*10^12 for a(4). - Christopher E. Thompson, Aug 06 2015

References

  • Dan Calistrate, Marc Paulhus and David Wolfe, On the lattice structure of finite games, in More Games of No Chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ., 42, Cambridge Univ. Press, Cambridge, 2002, pp. 25-30.
  • J. H. Conway, On Numbers and Games, Academic Press, NY, 1976.
  • Aaron N. Siegel, Combinatorial Game Theory, AMS Graduate Texts in Mathematics Vol 146 (2013), p. 158.

Crossrefs

Formula

a(n) = A125990(2*A114561(n)). - Antti Karttunen, Oct 18 2018

Extensions

Dean Hickerson and Robert Li found a(3) in 1974.

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

Views

Author

Antti Karttunen, Dec 18 2006

Keywords

Comments

Here we use the encoding described in A106486.

Examples

			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.
		

Crossrefs

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.

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

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

Number of terms of A126011 in range [0,2^n[.

Crossrefs

Formula

For all n, a(2*A114561(n)) = A065401(n).

A126012 A106486-encoding of the canonical representative of the combinatorial game with code n.

Original entry on oeis.org

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, 6, 32, 33, 32, 33, 36, 36, 36, 36, 32, 33, 32, 33, 36, 36, 36, 36, 48, 33, 48, 33, 36, 36, 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
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2006

Keywords

Examples

			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.
		

Crossrefs

A126011 gives the distinct terms (and also the records).
Showing 1-7 of 7 results.