A317367 - OEIS

%I #24 Jul 16 2019 23:49:36

%S 0,2,1,3,0,1,1,3,0,2,3,3,2,2,3,4,2,4,0,4,2,5,3,2,2,3,3,2,0,3,1,1,0,3,

%T 1,2,0,3,1,2,0,1,1,4,0,4,5,4,7,2,6,4,7,5,0,4,1,1,0,2,1,3,0,2,1,3,0,1,

%U 1,3,0,2,3,3,2,2,7,4,6,5,4,4,5,5,7,2,6,3,3,2,0,3,1,1,0,3,1,2,0,3,1,2,0,1,1

%N Sprague-Grundy values for the Node-Kayles game played on the scalloped graph where vertices that are 3 away from each other are connected.

%C a(n) is periodic with period 62 and final irregularity at n = 115 (proved).

%C A071461 gives Sprague-Grundy values for octal games .124 and .1241, where n=21 and n=49 differ.

%D E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 1, A K Peters, 2001.

%H Georg Fischer, <a href="/A317367/b317367.txt">Table of n, a(n) for n = 4..996</a> (first 300 terms from Eugene Fiorini)

%F Note that mex is the minimum excluded function and (+) is bitwise exclusive or. This is an obvious construction given the symmetry of the graph. We define a three-connected path graph P_{n}(3) as the path graph P_{n} with vertices labeled sequentially from 1 to n and additional edges E'={{i, j} : |i-j|=3}. The Sprague-Grundy value of P_{n}(3) is given by the function a(n).

%F The B-Variant of length n of P_{n}(3) (denoted P_{n}^{B}(3)) is exactly P_{n}(3) with one additional vertex -1 and one additional edge {-1,2}. The degree of vertex -1 is one. The Sprague-Grundy value of P_{n}^{B}(3) is given by the function b(n).

%F The C-Variant of length n of P_{n}(3) (denoted P_{n}^{C}(3)) is exactly P_{n}(3) with two additional vertices -1 and n+2 and additional edges {-1,2} and {n-1,n+2}. The degree of vertices -1 and n+2 is one. The Sprague-Grundy value of P_{n}^{C}(3) is given by the function c(n).

%F Then

%F a(n) = mex({b(i) (+) b(n-7-i) : -3 <= i <= ceiling(n/2)-4});

%F b(n) = mex({b(n-2)} union {c(i) (+) b(n-7-i): -3 <= i <= n-4});

%F c(n) = mex({c(n-2)} union {c(i) (+) c(n-7-i): -3 <= i <= ceiling(n/2)-4}). [Edited and extended by _Eugene Fiorini_, May 14 2019]

%t mex[S_] := Block[{i}, i=0; While[MemberQ[S,i], i=i+1]; i];

%t c={1}; Do[c = Append[c, mex[ Join[ {c[[n-2]]}, Table[ BitXor[ c[[i]], c[[n-3-i]] ], {i, Ceiling[(n-4)/2]} ] ] ] ], {n,2,1000}];

%t b={0}; Do[b = Append[b, mex[ Join[ {b[[n-2]]}, Table[ BitXor[ c[[i]], b[[n-3-i]] ], {i, n-4} ] ] ] ], {n,2,1000}];

%t a={}; Do[a = Append[a, mex[ Table[ BitXor[ b[[i]], b[[n-3-i]] ], {i, Ceiling[(n-4)/2]} ] ] ], {n,1000}]; (* _Eugene Fiorini_, May 14 2019 *)

%Y Cf. A071461.

%K nonn

%O 4,2

%A _Sierra Brown_, _Spencer Daugherty_, _Eugene Fiorini_, _Barbara Maldonado_, _Sean E. Rainville_, _Riley S. Waechter_, _Wing Hong Tony Wong_, Jul 26 2018

%E Edited by _N. J. A. Sloane_, Jan 30 2019

%E More terms from _Eugene Fiorini_, May 14 2019