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 1-6 of 6 results.

A119464 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z sorted by sum, ties broken by putting smallest value of x first, then y, then z; sequence gives x values.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 2, 3, 3, 0, 0, 1, 0, 2, 0, 1, 2, 3, 3, 0, 4, 0, 1, 4, 5, 5, 0, 2, 4, 6, 6, 0, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 7, 0, 0, 1, 0, 2, 0, 1, 2, 3, 3, 0, 4, 0, 1, 4, 5, 5, 0, 2, 4, 6, 6, 0, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 7, 0, 8, 0, 1, 8, 9, 9, 0, 2, 8, 10, 10, 0, 1, 2, 3, 3
Offset: 1

Views

Author

Joshua Zucker, May 21 2006

Keywords

Comments

Thanks to Ray Chandler for help in clarifying the relation between this sorted list and the one in A080593.

Examples

			The triples with sum <= 20 (this sequence is the first column) are:
0 0 0
0 1 1
0 2 2
0 3 3
1 2 3
0 4 4
0 5 5
1 4 5
0 6 6
2 4 6
0 7 7
1 6 7
2 5 7
3 4 7
3 5 6
0 8 8
0 9 9
1 8 9
0 10 10

Crossrefs

Cf. A119465, A119466 give the y and z values; A080593, A080594, A080595 give the same values sorted in a different way.

A080593 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives x values.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 4, 5, 6, 7, 0, 5, 4, 7, 6, 1, 0, 6, 7, 4, 5, 2, 3, 0, 7, 6, 5, 4, 3, 2, 1, 0, 0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 4, 5, 6, 7, 0, 5, 4, 7, 6, 1, 0, 6, 7, 4, 5, 2, 3, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 9, 10, 11, 12, 13, 14, 15, 0, 9, 8, 11, 10, 13
Offset: 0

Views

Author

N. J. A. Sloane, Feb 23 2003

Keywords

Comments

(x,y,z) is a losing position iff the mod-2 sum of the binary expansions of x,y,z (without carries) is 0.
In this sort the first few triples are: 0 0 0, 0 1 1, 0 2 2, 1 2 3, 0 3 3, 0 4 4, 1 4 5, 0 5 5, 2 4 6, 3 5 6, 0 6 6, 3 4 7, 2 5 7, 1 6 7, 0 7 7, 0 8 8, 1 8 9, 0 9 9, 2 8 10, 3 9 10, 0 10 10, 3 8 11, 2 9 11, 1 10 11, 0 11 11, 4 8 12, 5 9 12, 6 10 12, 7 11 12, 0 12 12. The 0,0,0 triple was added by Joshua Zucker.

References

  • I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

Crossrefs

Cf. A080594, A080595.
A119464, A119465, A119466 give the same terms as these sequences but sorted in a different order (by sum rather than by value of z).

Extensions

Corrected and extended by John W. Layman, Oct 22 2003

A080594 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives y values.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 7, 8, 8, 9, 8, 9, 10, 8, 9, 10, 11, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 16, 17, 18, 16, 17, 18, 19, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 21, 16, 17, 18, 19, 20, 21, 22, 16
Offset: 0

Views

Author

N. J. A. Sloane, Feb 23 2003

Keywords

Comments

(x,y,z) is a losing position iff the mod-2 sum of the binary expansions of x,y,z (without carries) is 0.
In this sort the first few triples are: 0 0 0, 0 1 1, 0 2 2, 1 2 3, 0 3 3, 0 4 4, 1 4 5, 0 5 5, 2 4 6, 3 5 6, 0 6 6, 3 4 7, 2 5 7, 1 6 7, 0 7 7, 0 8 8, 1 8 9, 0 9 9, 2 8 10, 3 9 10, 0 10 10, 3 8 11, 2 9 11, 1 10 11, 0 11 11, 4 8 12, 5 9 12, 6 10 12, 7 11 12, 0 12 12. The 0,0,0 triple was added by Joshua Zucker.

References

  • I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

Crossrefs

Cf. A080593, A080595.
A119464, A119465, A119466 give the same terms as these sequences but sorted in a different order (by sum rather than by value of z).

Extensions

More terms from John W. Layman, Oct 22 2003

A080595 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives z values.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 22
Offset: 0

Views

Author

N. J. A. Sloane, Feb 23 2003

Keywords

Comments

(x,y,z) is a losing position iff the mod-2 sum of the binary expansions of x,y,z (without carries) is 0.
In this sort the first few triples are: 0 0 0, 0 1 1, 0 2 2, 1 2 3, 0 3 3, 0 4 4, 1 4 5, 0 5 5, 2 4 6, 3 5 6, 0 6 6, 3 4 7, 2 5 7, 1 6 7, 0 7 7, 0 8 8, 1 8 9, 0 9 9, 2 8 10, 3 9 10, 0 10 10, 3 8 11, 2 9 11, 1 10 11, 0 11 11, 4 8 12, 5 9 12, 6 10 12, 7 11 12, 0 12 12. The 0,0,0 triple was added by Joshua Zucker.

References

  • I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

Crossrefs

Cf. A080594, A080593.
A119464, A119465, A119466 give the same terms as these sequences but sorted in a different order (by sum rather than by value of z).

A119465 Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z sorted by sum, ties broken by putting smallest value of x first, then y, then z; sequence gives y values.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 5, 4, 6, 4, 7, 6, 5, 4, 5, 8, 9, 8, 10, 8, 11, 10, 9, 8, 9, 12, 8, 13, 12, 9, 8, 9, 14, 12, 10, 8, 10, 15, 14, 13, 12, 13, 11, 10, 11, 9, 11, 8, 9, 10, 11, 16, 17, 16, 18, 16, 19, 18, 17, 16, 17, 20, 16, 21, 20, 17, 16, 17, 22, 20, 18, 16, 18, 23, 22, 21, 20, 21, 19, 18
Offset: 1

Views

Author

Joshua Zucker, May 21 2006

Keywords

Comments

Thanks to Ray Chandler for help in clarifying the relation between this sorted list and the one in A080594.

Examples

			The triples with sum <= 20 (this sequence is the second column) are:
0 0 0
0 1 1
0 2 2
0 3 3
1 2 3
0 4 4
0 5 5
1 4 5
0 6 6
2 4 6
0 7 7
1 6 7
2 5 7
3 4 7
3 5 6
0 8 8
0 9 9
1 8 9
0 10 10

Crossrefs

A119464, A119466 give the x and z values; A080593, A080594, A080595 give the same values sorted in a different way.

A190588 Losing positions in Nim (misere) with up to 9 stones on each heap.

Original entry on oeis.org

1, 22, 33, 44, 55, 66, 77, 88, 99, 111, 123, 145, 167, 189, 246, 257, 347, 356, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1247, 1256, 1346, 1357, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 2345, 2367, 2389, 3333, 3344, 3355, 3366, 3377, 3388, 3399, 4444, 4455, 4466, 4477, 4488, 4499, 4567, 4589, 5555, 5566, 5577, 5588, 5599, 6666, 6677, 6688, 6699, 6789, 7777, 7788, 7799, 8888, 8899, 9999, 11111
Offset: 1

Views

Author

Sergio Pimentel, May 13 2011

Keywords

Comments

In the game of Nim (misere) the player who takes the last stone loses. This sequence is a subsequence of A009994 (numbers with digits in nondecreasing order).

Examples

			Example: a(2) is 22 because it represents a losing position with two heaps with two stones in each. In the game of Nim (misere) the player who takes the last stone loses. If Player A takes one stone from heap 1, then Player B takes two from heap 2, leaving Player A to pick the last stone and lose the game.

Links

  • Wikipedia, Nim

Crossrefs

Cf. A009994, A119456, A119464, A119466.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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