%I
%S 0,1,5,9,15,21,25,29,35,39,43,55,59,63,73,77,89,93,97,107,111,123,127,
%T 131,141,145,157,161,165,175,179,191,195,199,209,213,225,229,233
%N The values of N for which the 1 X N dominocovering game is a second player win.
%C In a dominocovering game, the players take turns placing dominoes (1 X 2 rectangles) on the board (here a 1 X N rectangle), such that the position of the dominoes is integer, and no two dominoes overlap. The loser is the first player unable to move.
%C The optimal strategies for this game can be determined by computing the nimvalue nv(N) of each board 1 X N:
%C nv(0) = 0
%C nv(1) = 0
%C nv(N+2) = least nonnegative integer not in {nimsum(nv(k),nv(Nk)) : k <= N}
%C (where nimsum(a,b) is the bitwise xor of a and b).
%C The second player wins a game iff its nimvalue is 0.
%H Charles R Greathouse IV, <a href="/A215721/b215721.txt">Table of n, a(n) for n = 1..10000</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 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,1).
%F For n > 14, a(n) = a(n5) + 34.  _Rainer Rosenthal_ and _Charles R Greathouse IV_, Sep 20 2012
%F G.f.: x^2*(6*x^13  2*x^12 + 8*x^10  2*x^9  2*x^8 + 2*x^7 + 3*x^5 + 6*x^4 + 6*x^3 + 4*x^2 + 4*x + 1) / ((x  1)^2*(x^4 + x^3 + x^2 + x + 1)).  _Colin Barker_, Jan 26 2013
%e N=1: the first player is unable to move (second player win).
%e N=2,3: any move by the first player renders the second player unable to move (first player win).
%e N=4: the first player can win by covering the two central squares (first player win).
%e N=5: any move by the first player has a final countermove by the second (second player win).
%o (Python)
%o import numpy as np
%o N = np.array([0,0])
%o U = np.arange(1000)
%o for i in U:
%o ..N = np.append(N, np.setdiff1d(U,np.bitwise_xor(N[:1],N[2::1])).min())
%o print list(*np.where(N==0))
%o (PARI) a(n)=if(n<10,[0,1,5,9,15,21,25,29,35][n],n\5*34[29,25,13,9,5][n%5+1]) \\ _Charles R Greathouse IV_, Aug 24 2012
%o (PARI) list(lim)=my(v=vector(lim\1+1),u);for(n=0,#v3,u=vecsort(vector(n\2+1,k,bitxor(v[k],v[nk+2])),,8);for(i=0,#u1,if(u[i+1]!=i,v[n+3]=i;next(2));v[n+3]=#u));for(i=0,#v1,v[i+1]=if(v[i+1],1,i)); vecsort(v,,8) \\ _Charles R Greathouse IV_, Aug 24 2012
%Y The nimvalues are A002187(n1).
%K nonn,easy
%O 1,3
%A _Yuval Gabay_, Aug 22 2012
