A374920 -id:A374920 - cp's OEIS frontend

A374810 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+2) = v(1)v(2)...v(k+2) with a (1^k,0,1)-weight assignment is a second-player winning game.

1, 6, 7, 12, 13, 18, 23, 24, 38, 39, 44, 45, 50, 51, 56, 62, 77, 115, 121, 153, 312, 333, 350, 427, 553, 554, 579

Offset: 1

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

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 legal only provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G = P(k+2), a (1^k, 0, 1)-weight assignment is one in which vertex v(k+1) is assigned weight 0 and all remaining vertices are assigned weight 1.

			For n = 6, the {0,1}-Toggle game on P(8) with a (1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
For n = 12, the {0,1}-Toggle game on P(14) with a (1,1,1,1,1,1,1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.

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

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.

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374910, A374920

Python
```
# See Cohen link.
```

A374910 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+4) = v(1)v(2)...v(k+4) with a (1^k,0,1,0,1)-weight assignment is a second-player winning game.

Original entry on oeis.org

1, 25, 26, 31, 32, 37, 38, 63, 64, 69, 70, 76, 101, 102, 139, 145, 177, 189, 215, 235, 252, 253, 267, 284, 290, 305, 311, 328, 360, 668

Offset: 1

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

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+4), a (1^k,0,1,0,1)-weight assignment is one in which vertices v(k+1) and v(k+3) are assigned weight 0 and all remaining vertices are assigned weight 1.
The path P(k+4m) 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.

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

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.

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374810, A374920.

Python
```
# See Cohen link.
```

cp's OEIS Frontend

A374810 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+2) = v(1)v(2)...v(k+2) with a (1^k,0,1)-weight assignment is a second-player winning game.

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