A374920 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+6) = v(1)v(2)...v(k+6) with a (1^k,0,1,0,1,0,1)-weight assignment is a second-player winning game.

1, 6, 7, 12, 13, 18, 23, 24, 39, 44, 45, 50, 51, 57, 62, 77, 115, 281, 319, 350, 389

Offset

1,2

Comments

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.

A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.

In the special case G=P(k+6), a (1^k,0,1,0,1,0,1)-weight assignment is one in which vertices v(k+1), v(k+3), and v(k+5) are assigned weight 0 and all remaining vertices are assigned weight 1.

The path P(k+4m+2) where vertices v(k+1), v(k+3), ..., v(k+4m+1) are assigned weight 0 and all remaining vertices are assigned weight 1 will have the same Grundy numbers as G.

References

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Table of n, a(n) for n=1..21.

K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, Generalized Impartial Two-player Pebbling Games on K3 and C4, Journal of Integer Sequences, 27(5), 2024.

Matthew Cohen, Python

E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.

Prog

(Python) # See Cohen link.

Crossrefs

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374810, A374910

Sequence in context: A327313 A047225 A374810 * A191337 A352090 A276089

Adjacent sequences: A374917 A374918 A374919 * A374924 A374925 A374926

Keyword

nonn,more

Author

Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024

Status

approved