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%I M0410 N0156 #41 Jan 29 2022 01:13:29
%S 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,
%T 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,
%U 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
%N Sprague-Grundy values for the game of Kayles (octal games .77 and .771).
%C Octal games 4.4, 4.41, 4.42, 4.43, 4.6, 4.61, 4.62 and 4.63 have values a(n-1).
%C "The periodicity was first proved by _R. K. Guy_ in 1949, the sequence necessarily being calculated by hand." [Beasley].
%D John D. Beasley, The Mathematics of Games, Dover Publ., Mineola, NY 2006, page 111.
%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 91.
%D 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.)
%D J. H. Conway, On Numbers and Games, Second Edition. A K Peters, Ltd, 2001, p. 128.
%D R. K. Guy, "Anyone for Twopins?", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
%D Guy, R. K. and Smith, C. A. B.; The G-values of various games. Proc. Cambridge Philos. Soc. 52 (1956), 514-526.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Sierra Brown, Spencer Daugherty, Eugene Fiorini, Barbara Maldonado, Diego Manzano-Ruiz, Sean Rainville, Riley Waechter, and Tony W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Wong/wong24.html">Nimber Sequences of Node-Kayles Games</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.5.
%H Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/octal_sparse.html">Octal games</a>
%H R. K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>
%H R. K. Guy, <a href="/A005251/a005251_1.pdf">Anyone for Twopins?</a>, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,1).
%F 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.
%Y Cf. A071074, A071434.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 08 2001
%E Edited by _Christian G. Bower_, Oct 22 2002
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