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 1-9 of 9 results.

A037142 Erroneous version of A047995: Number of impartial misere games born on or before day n.

Original entry on oeis.org

1, 4, 22, 4171780
Offset: 0

Views

Author

Jeffrey Harris (alvinharris(AT)home.com)

Keywords

Comments

Apparently a conflation of A047995 (impartial games, misere play) and A065401 (partisan games, normal play) based on the (surely?) accidental appearance of 22 in both sequences. - Christopher E. Thompson, Nov 17 2015

Crossrefs

Cf. A047995, A065401.

Formula

gamma(0) = 0.149027998351785...; gamma(n+1) = (2^gamma(n)); f(n) = ceiling(gamma(n)).

Extensions

Next term = (2^4171780)-(2^2095104)-(3*2^2094593)-(2^2094081)-(3*2^2091522)-(2^2088960)-(3*2^2088448)-(2^2087937)-(2^2086912)-(2^2086657)-(2^2086401)-(2^2086145)-(2^2085888)-(2^2079234)+(2^1960962)+21.
Probably this is an incorrect version of A047995. - N. J. A. Sloane

A065402 Number of normal play partisan games born on day n.

Original entry on oeis.org

1, 3, 18, 1452
Offset: 0

Views

Author

R. K. Guy, Nov 23 2001

Keywords

Comments

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

Examples

			Day 0: 0; day 1: 1 -1 *; day 2, 18: 2 -2 1/2 -1/2 1* -1* +/-1 up upstar down downstar *2 1|* 1|0 1|0,* *|-1 0|-1 0,*|-1.

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.

Crossrefs

Cf. A065401, A065407, A047995, A037142.

A065407 Number of levels in lattice formed by normal play partisan games born on day n.

Original entry on oeis.org

1, 3, 9, 45, 2949
Offset: 0

Views

Author

R. K. Guy, Nov 23 2001

Keywords

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.

Crossrefs

Cf. A065401, A065402, A047995, A037142. Equals 2*A065402(n-1) + 1.

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[.

Links

Crossrefs

Cf. A065401, A125999, A126010, A126011, A126012, A320006.

Formula

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

A260967 Number of infinitesimal normal play partisan games born on or before day n.

Original entry on oeis.org

1, 2, 7, 292
Offset: 0

Views

Author

Christopher E. Thompson, Aug 06 2015

Keywords

Comments

A game g is infinitesimal if -x < g < x for all positive numbers x, or equivalently if its left and right stops are both 0.

References

  • Aaron N. Siegel, Combinatorial Game Theory, AMS Graduate Texts in Mathematics Vol 146 (2013), p. 158.

Crossrefs

Cf. A065401 (all games), A260968 (all-small games), A260969 (reduced games).

A260969 Number of reduced normal play partisan games born on or before day n.

Original entry on oeis.org

1, 3, 10, 61, 15845
Offset: 0

Views

Author

Christopher E. Thompson, Aug 06 2015

Keywords

Comments

A game is reduced if it is the simplest among the set of games with values infinitesimally different from it.
The values up to a(4) are due to Aaron Siegel, computed using cgsuite.

References

  • Aaron N. Siegel, Combinatorial Game Theory, AMS Graduate Texts in Mathematics Vol 146 (2013), p. 158.

Links

Crossrefs

Cf. A065401 (all games), A260967 (infinitesimal games).

A260968 Number of all-small normal play partisan games born on or before day n.

Original entry on oeis.org

1, 2, 7, 67, 534483
Offset: 0

Views

Author

Christopher E. Thompson, Aug 06 2015

Keywords

Comments

A game is all-small if it and all its followers other than 0 have options for both players, or equivalently (under normal play rules) if it and all its followers are infinitesimal. "All-small" is the traditional term, although Aaron Siegel prefers "dicotic", attributed to Michael Weimerskirch.
The all-small games born by day n form a distributive lattice if additional minimal and maximal elements are added.
The values up to a(4) are due to Aaron Siegel, computed using cgsuite.

References

  • Aaron N. Siegel, Combinatorial Game Theory, AMS Graduate Texts in Mathematics Vol 146 (2013), p. 158.

Links

  • David Wolfe, On day n, pp. 125-131 in Games of No Chance 3, MSRI Publications 56, Cambridge, 2009.

Crossrefs

Cf. A065401 (all games), A260967 (infinitesimal games).

A260970 Number of hereditarily transitive normal play partisan games born on or before day n.

Original entry on oeis.org

1, 4, 18, 176, 11363
Offset: 0

Views

Author

Christopher E. Thompson, Aug 07 2015

Keywords

Comments

A game is transitive if any position reached by any number of consecutive moves by one player can be reached in a single move by that player. It is hereditarily transitive if it and all its followers are transitive.
The hereditarily transitive games born by day n form a distributive lattice whose Hasse diagram is planar. It is conjectured (known for n<=3) that the number of antichains in this lattice is 2^A000372(n)-2.
Aaron Siegel attributes the values up to a(3) to Angela Siegel, and a(4) to Neil McKay.

References

  • Aaron N. Siegel, Combinatorial Game Theory, AMS Graduate Texts in Mathematics Vol 146 (2013), p. 158.

Crossrefs

Cf. A065401 (all games), also A000372 for antichain conjecture.

A368588 Number of misère-play left dead end games born by day n.

Original entry on oeis.org

1, 2, 4, 10, 52, 21278
Offset: 0

Views

Author

Aaron N. Siegel, Dec 31 2023

Keywords

Comments

A partizan combinatorial game G is a left dead end if no subposition of G has any left options. In normal play, every left dead end is equal to a nonpositive integer. In misère play, the left dead ends have a more intricate structure; this sequence counts the misère-inequivalent left dead ends with birthday <= n.

References

  • E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.

Links

Crossrefs

Cf. A065401, A047995, A260969.
Showing 1-9 of 9 results.

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

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