A071434 -id:A071434 - cp's OEIS frontend

A002186 Sprague-Grundy values for the game of Kayles (octal games .77 and .771).

0, 1, 2, 3, 1, 4, 3, 2, 1, 4, 2, 6, 4, 1, 2, 7, 1, 4, 3, 2, 1, 4, 6, 7, 4, 1, 2, 8, 5, 4, 7, 2, 1, 8, 6, 7, 4, 1, 2, 3, 1, 4, 7, 2, 1, 8, 2, 7, 4, 1, 2, 8, 1, 4, 7, 2, 1, 4, 2, 7, 4, 1, 2, 8, 1, 4, 7, 2, 1, 8, 6, 7, 4, 1, 2, 8, 1, 4, 7, 2, 1, 8, 2, 7, 4, 1, 2, 8, 1, 4, 7, 2, 1, 8, 2, 7, 4, 1, 2, 8, 1, 4, 7, 2, 1

Offset: 0

N. J. A. Sloane

Octal games 4.4, 4.41, 4.42, 4.43, 4.6, 4.61, 4.62 and 4.63 have values a(n-1).

"The periodicity was first proved by R. K. Guy in 1949, the sequence necessarily being calculated by hand." [Beasley].

John D. Beasley, The Mathematics of Games, Dover Publ., Mineola, NY 2006, page 111.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 91.
Calkin, Neil J.; James, Kevin; Janoski, Janine E.; Leggett, Sarah; Richards, Bryce; Sitaraman, Nathan; and Thomas, Stephanie M.; Computing strategies for graphical Nim, in Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 171-185. (See page 174.)
J. H. Conway, On Numbers and Games, Second Edition. A K Peters, Ltd, 2001, p. 128.
R. K. Guy, "Anyone for Twopins?", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
Guy, R. K. and Smith, C. A. B.; The G-values of various games. Proc. Cambridge Philos. Soc. 52 (1956), 514-526.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sierra Brown, Spencer Daugherty, Eugene Fiorini, Barbara Maldonado, Diego Manzano-Ruiz, Sean Rainville, Riley Waechter, and Tony W. H. Wong, Nimber Sequences of Node-Kayles Games, J. Int. Seq., Vol. 23 (2020), Article 20.3.5.
Achim Flammenkamp, Octal games
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A071074, A071434.

From n=71 on, the sequence is periodic with period 12. The only exceptions are n=0, 3, 6, 9, 11, 15, 18, 21, 22, 28, 34, 39, 57 and 70.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 08 2001

Edited by Christian G. Bower, Oct 22 2002

A126110 Number of misere quotients of order 2n.

Original entry on oeis.org

1, 0, 1, 1, 1, 6, 9, 50, 211

Offset: 1

Jonathan Vos Post, Mar 05 2007

Siegel's abstract: "A bipartite monoid is a commutative monoid Q together with an identified subset P subset of Q. In this paper we study a class of bipartite monoids, known as misere quotients, that are naturally associated to impartial combinatorial games. We introduce a structure theory for misere quotients with |P| = 2 and give a complete classification of all such quotients up to isomorphism. One consequence is that if |P| = 2 and Q is finite, then |Q| = 2^n+2 or 2^n+4. We then develop computational techniques for enumerating misere quotients of small order and apply them to count the number of non-isomorphic quotients of order at most 18. We also include a manual proof that there is exactly one quotient of order 8." [Quotation corrected by Thane Plambeck, Jul 08 2014]

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 89 and 102.
J. H. Conway, On Numbers and Games, Second Edition. A. K. Peters, Ltd, 2001, p. 128.
T. E. Plambeck, Advances in Losing, in M. Albert and M. J. Nowakowski, eds., Games of No Chance 3, Cambridge University Press, forthcoming.

Achim Flammenkamp, Sparse- and Common-Positions of Sprague-Grundy Values of Octal-Games
Aaron N. Siegel, The structure and classification of misere quotients, figure 1, p. 3, 2 Mar 2007.

Cf. A071074, A071434.

cp's OEIS Frontend

A002186 Sprague-Grundy values for the game of Kayles (octal games .77 and .771).

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