A215721 The values of N for which the 1 X N domino-covering game is a second player win.

0, 1, 5, 9, 15, 21, 25, 29, 35, 39, 43, 55, 59, 63, 73, 77, 89, 93, 97, 107, 111, 123, 127, 131, 141, 145, 157, 161, 165, 175, 179, 191, 195, 199, 209, 213, 225, 229, 233, 243, 247, 259, 263, 267, 277, 281, 293, 297, 301, 311, 315, 327, 331, 335, 345, 349, 361

Offset

1,3

Comments

In a domino-covering 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 make a move.

The optimal strategies for this game can be determined by computing the nim-value nv(N) of each board 1 X N:

nv(0) = 0

nv(1) = 0

nv(N+2) = least nonnegative integer not in {nimsum(nv(k),nv(N-k)) : k <= N}

(where nimsum(a,b) is the bitwise xor of a and b).

The second player wins a game iff its nim-value is 0.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, and Kevin Zhang Kevin Zhao, PRIMES STEP Plays Games, arXiv:1707.07201 [math.CO], 2017, Section 8.

Stephan Brumme, Project Euler.net Problem 306, (2017).

Randal E. Bryant and Marijn J. H. Heule, Dual Proof Generation for Quantified Boolean Formulas with a BDD-Based Solver, Carnegie Mellon Univ. (2021).

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Formula

For n > 14, a(n) = a(n-5) + 34. - Rainer Rosenthal and Charles R Greathouse IV, Sep 20 2012

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

Example

N=1: the first player is unable to move (second player win).
N=2,3: any move by the first player renders the second player unable to move (first player win).
N=4: the first player can win by covering the two central squares (first player win).
N=5: any move by the first player has a final counter-move by the second (second player win).

Mathematica

Rest@ CoefficientList[Series[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)), {x, 0, 39}], x] (* Michael De Vlieger, Aug 18 2021 *)

Prog

(Python)
import numpy as np
N = np.array([0, 0])
U = np.arange(1000)
for i in U:
    N = np.append(N, np.setdiff1d(U, np.bitwise_xor(N[:-1], N[-2::-1])).min())
print(list(*np.where(N==0)))
(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
(PARI) list(lim)=my(v=vector(lim\1+1), u); for(n=0, #v-3, u=vecsort(vector(n\2+1, k, bitxor(v[k], v[n-k+2])), , 8); for(i=0, #u-1, if(u[i+1]!=i, v[n+3]=i; next(2)); v[n+3]=#u)); for(i=0, #v-1, v[i+1]=if(v[i+1], 1, i)); vecsort(v, , 8) \\ Charles R Greathouse IV, Aug 24 2012

Crossrefs

The nim-values are A002187(n-1).

Sequence in context: A315072 A315073 A315074 * A315075 A315076 A350472

Adjacent sequences: A215718 A215719 A215720 * A215722 A215723 A215724

Keyword

nonn,easy

Author

Yuval Gabay, Aug 22 2012

Status

approved