A361517 The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.

3, 4, 5, 11, 17, 27, 35, 37, 49, 59, 69, 81, 91, 103, 115, 123, 135, 137, 167, 175, 189, 199, 207, 287, 295, 307, 361, 1051, 2507, 2757, 2917, 3057, 3081, 7255, 7361, 7871, 16173

Offset

3,1

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 weights 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=GP(n,2), a (1,0)-weight assignment is one in which each vertex of the outer polygon is assigned weight 1 and each vertex of the inner polygon(s) is assigned weight 0.

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=3..39.

Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob Porter, and Andrew Woldar, On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out, arXiv:2411.08247 [math.CO], 2024. See p. 17.

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.

Katherine Levandosky, CGSuite Program.

Example

For n = 3, the {0,1}-Toggle game on GP(3,2) with a (1,0)-weight assignment is a next-player winning game.
For n = 5, the {0,1}-Toggle game on GP(5,2) with a (1,0)-weight assignment is a next-player winning game.

Prog

(CGSuite) # See Levandosky link

Crossrefs

Cf. A071426, A340631, A346197, A346401, A346637.

Sequence in context: A362665 A293713 A280255 * A276542 A318077 A074221

Adjacent sequences: A361514 A361515 A361516 * A361518 A361519 A361520

Keyword

nonn,more

Author

Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob K. Porter and Andrew Woldar, Mar 14 2023

Status

approved