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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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].

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 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).

Links

• 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).

Crossrefs

Cf. A071074, A071434.

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.

Extensions

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

Edited by Christian G. Bower, Oct 22 2002