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%I M0025 N0007 #27 Jul 06 2020 16:55:26
%S 0,1,1,2,0,3,1,1,0,3,3,2,2,4,0,5,2,2,3,3,0,1,1,3,0,2,1,1,0,4,5,2,7,4,
%T 0,1,1,2,0,3,1,1,0,3,3,2,2,4,4,5,5,2,3,3,0,1,1,3,0,2,1,1,0,4,5,3,7,4,
%U 8,1,1,2,0,3,1,1,0,3,3,2,2,4,4,5,5,9,3,3,0,1,1,3,0,2,1,1,0,4,5,3,7,4,8,1,1,2,0,3,1,1,0,3,3,2,2,4,4,5,5,9,3,3,0,1,1,3,0,2,1,1,0,4
%N Sprague-Grundy values for Dawson's Chess (octal game .137).
%C Octal game .07 (Dawson's Kayles) has values a(n-1). Octal games .4, .401, .402, .403, .42, .421, .422 and .423 have values a(n-2).
%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 89 and 102.
%D R. K. Guy and C. A. B. Smith, 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 Reinhard Zumkeller, <a href="/A002187/b002187.txt">Table of n, a(n) for n = 0..1000</a>
%H Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, Kevin Zhang Kevin Zhao, <a href="https://arxiv.org/abs/1707.07201">PRIMES STEP Plays Games</a>, arXiv:1707.07201 [math.CO], 2017, Section 8.
%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="/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]
%F Has period 34 with the only exceptions at n=0, 14, 16, 17, 31, 34 and 51.
%o (Haskell)
%o a002187 n = a002187_list !! n
%o a002187_list = tail g where
%o g = 0 : 0 : [mex [xor (g !! (a + 1)) (g !! (n - a - 2)) |
%o a <- [-1 .. n - 2]] | n <- [1 ..]]
%o xor 0 0 = 0
%o xor x y = let ((q,r), (s,t)) = (divMod x 2, divMod y 2)
%o in (if r == t then 0 else 1) + 2 * xor q s
%o mex xs = head [x | x <- [0..], not (elem x xs)]
%o -- Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 08 2005; edited by _Reinhard Zumkeller_, Dec 16 2013
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Oct 22 2002
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