

A002186


SpragueGrundy values for the game of Kayles (octal games .77 and .771).
(Formerly M0410 N0156)


28



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
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OFFSET

0,3


COMMENTS

Octal games 4.4, 4.41, 4.42, 4.43, 4.6, 4.61, 4.62 and 4.63 have values a(n1).
"The periodicity was first proved by R. K. Guy in 1949, the sequence necessarily being calculated by hand." [Beasley].


REFERENCES

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 FortyFirst Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 171185. (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. 215.
Guy, R. K. and Smith, C. A. B.; The Gvalues of various games. Proc. Cambridge Philos. Soc. 52 (1956), 514526.
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).


LINKS

Table of n, a(n) for n=0..104.
Sierra Brown, Spencer Daugherty, Eugene Fiorini, Barbara Maldonado, Diego ManzanoRuiz, Sean Rainville, Riley Waechter, and Tony W. H. Wong, Nimber Sequences of NodeKayles Games, J. Int. Seq., Vol. 23 (2020), Article 20.3.5.
Achim Flammenkamp, Octal games
R. K. Guy, Letters to N. J. A. Sloane, JuneAugust 1968
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 215. [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).


FORMULA

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.


CROSSREFS

Cf. A071074, A071434.
Sequence in context: A242360 A299927 A006021 * A125936 A243614 A200942
Adjacent sequences: A002183 A002184 A002185 * A002187 A002188 A002189


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 08 2001
Edited by Christian G. Bower, Oct 22 2002


STATUS

approved



