A364489 Values of n for which the Sprague-Grundy value of Heat-Charge Toggle on an (n+2)-vertex path with initial weights -1,1^n,-1 is evil for odd n or odious for even n.

1, 4, 6, 9, 14, 22, 27, 30, 35, 41, 58, 59, 72, 84, 87, 89, 103, 105, 108, 124, 129, 141, 171, 258, 284, 407, 458, 11770548, 25146268, 27690032, 41693544, 55788270, 74838555, 86120064, 89811321, 95580294, 119784327, 139336981, 158776090, 160066751, 161102638, 181691114, 186919128

Offset: 1

Author

Jean-Pierre Appel, Patrick G. Cesarz, Djeneba Diop, Eugene Fiorini, Nathan Hurtig, Andrew Woldar, Jul 26 2023

Keywords

• nonn

Comments

Heat-Charge Toggle is an impartial two-player game on a finite simple graph, where each vertex is assigned a weight of -1, 0, or 1.

A Heat-Charge Toggle move consists of selecting a vertex of weight 1 and switching its weight to 0, as well as switching the sign of each of its neighbors: changing 1 to -1, -1 to 1, and keeping 0 at 0.

We additionally only allow moves that strictly decrease the sum of all weights.

The two vertices of degree one have initial weights of -1, while vertices of degree two have an initial weight of 1.

Examples

			For n = 4, the Sprague-Grundy value for a 6-vertex path is 2.
Note that n = 4 is even and 2 is odious (see A000069).
		

References

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

Links

• Nathan Hurtig, C++ program to compute values

Crossrefs

Cf. A000069, A001969, A361517, A363934. Paths with Sprague-Grundy values A364503.