A002187 Sprague-Grundy values for Dawson's Chess (octal game .137).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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).

References

• E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 89 and 102.
• R. K. Guy and C. A. B. Smith, 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

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, Kevin Zhang Kevin Zhao, PRIMES STEP Plays Games, arXiv:1707.07201 [math.CO], 2017, Section 8.
• 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, 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]

Programs

Haskell

a002187 n = a002187_list !! n
a002187_list = tail g where
   g = 0 : 0 : [mex [xor (g !! (a + 1)) (g !! (n - a - 2)) |
                     a <- [-1 .. n - 2]] | n <- [1 ..]]
   xor 0 0 = 0
   xor x y = let ((q,r), (s,t)) = (divMod x 2, divMod y 2)
              in (if r == t then 0 else 1) + 2 * xor q s
   mex xs = head [x | x <- [0..], not (elem x xs)]
-- Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 08 2005; edited by Reinhard Zumkeller, Dec 16 2013

Formula

Has period 34 with the only exceptions at n=0, 14, 16, 17, 31, 34 and 51.

Extensions

Edited by Christian G. Bower, Oct 22 2002