Patrick G. Cesarz - 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.
```

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.

Original entry on oeis.org

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

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+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.

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, A374910

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

A363894 Number of weakly connected components of a halved addsub configuration graph with respect to integers mod n over a path with two vertices.

Original entry on oeis.org

1, 2, 1, 8, 2, 3, 1, 5, 8, 4, 2, 18, 3, 33, 1, 19, 5, 6, 8, 20, 4, 7, 2, 39, 18, 14, 3, 32, 33, 25, 1, 29, 19, 58, 5, 42, 6, 75, 8, 91, 20, 34, 4, 108, 7, 13, 2, 17, 39, 164, 18, 58, 14, 83, 3, 47, 32, 16, 33, 66, 25, 167, 1, 365, 29, 18, 19, 56, 58, 19, 5

Offset: 2

Patrick G. Cesarz, Eugene Fiorini, Charles Gong, Kyle A. Kelley, Philip Thomas, and Andrew Woldar, Jul 23 2023

nonn

The addsub game is played on a path with two vertices {u,v}. We define a configuration of the integers mod n on {u,v} by assigning weights wt(u) and wt(v).
An addsub move from u to v is a reassignment of weights given by wt(u) -> wt(u) - wt(v) (mod n) and wt(v) -> wt(u) + wt(v) (mod n). An addsub move from v to u (i.e. the backward move) is defined analogously.
The halved addsub configuration graph is the directed subgraph of the addsub configuration graph restricted to the forward move only: wt(u) -> wt(u) - wt(v) (mod n) and wt(v) -> wt(u) + wt(v) (mod n).

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

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).
E. Fiorini, M. Lind, and A. Woldar, On Properties of Pebble Assignment Graphs, Graphs and Combinatorics, 38(2) (2022), 45.
E. Fiorini, G. Johnston, M. Lind, A. Woldar, and T. W. H. Wong, Cycles and Girth in Pebble Assignment Graphs, Graphs and Combinatorics, 38(5) (2022), 154.

Cf. A340631, A346197, A346401, A363893.

Upto=25;
Table[
  VertexSet:={};
  EdgeSet:={};
  (* Compute configuration graph for integers mod n *)
  Do[
    Do[AppendTo[VertexSet,{i,j}];
      AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[i-j,n],Mod[i+j,n]}],
      {j,0,n-1}],
    {i,0,n-1}];
  (* Print n-th term *)
  Length[WeaklyConnectedComponents[Graph[VertexSet,EdgeSet]]],
  {n,2,Upto}]

A363893 Number of weakly connected components of an addsub configuration graph with respect to integers mod n over a path with two vertices.

Original entry on oeis.org

1, 2, 1, 4, 2, 3, 1, 5, 4, 4, 2, 6, 3, 11, 1, 11, 5, 6, 4, 12, 4, 7, 2, 13, 6, 14, 3, 10, 11, 25, 1, 29, 11, 18, 5, 12, 6, 21, 4, 25, 12, 34, 4, 32, 7, 13, 2, 17, 13, 48, 6, 16, 14, 25, 3, 47, 10, 16, 11, 18, 25, 87, 1, 95, 29, 18, 11, 32, 18, 19, 5

Offset: 2

Patrick G. Cesarz, Eugene Fiorini, Charles Gong, Kyle A. Kelley, Philip Thomas, and Andrew Woldar, Jun 26 2023

nonn

The addsub game is played on a path with two vertices {u,v}. We define a configuration of the integers mod n on {u,v} by assigning weights wt(u) and wt(v).
An addsub move from u to v is a reassignment of weights given by wt(u) -> wt(u) - wt(v) (mod n) and wt(v) -> wt(u) + wt(v) (mod n). An addsub move from v to u is defined analogously.
The addsub configuration graph with respect to the integers mod n over {u,v} is the directed graph in which each node corresponds to a configuration (wt(u),wt(v)) and a directed edge from a configuration to the resulting configuration is attainable via a single addsub move.

			For n=3, the (u,v) sequence of addsub moves forms the directed cycle (0,1)->(2,1)->(1,0)->(1,1)->(0,2)->(1,2)->(2,0)->(2,2)->(0,1). The (v,u) sequence of addsub moves forms the directed cycle (0,1)->(1,1)->(2,0)->(2,1)->(0,2)->(2,2)->(1,0)->(1,2)->(0,1). These two directed cycles form one weakly connected component. The isolated vertex (0,0) is a loop and forms the second weakly connected component. Therefore, a(3)=2.

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

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).
E. Fiorini, M. Lind, and A. Woldar, On Properties of Pebble Assignment Graphs, Graphs and Combinatorics, 38(2) (2022), 45.
E. Fiorini, G. Johnston, M. Lind, A. Woldar, and T. W. H. Wong, Cycles and Girth in Pebble Assignment Graphs, Graphs and Combinatorics, 38(5) (2022), 154.

Cf. A340631, A346197, A346401.

Upto=25;
Table[
  VertexSet:={};
  EdgeSet:={};
  (* Compute configuration graph for integers mod n *)
  Do[
    Do[AppendTo[VertexSet,{i,j}];
      AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[i-j,n],Mod[i+j,n]}];
      AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[j+i,n],Mod[j-i,n]}],
      {j,0,n-1}],
    {i,0,n-1}];
  (* Print n-th term *)
  Length[WeaklyConnectedComponents[Graph[VertexSet,EdgeSet]]],
  {n,2,Upto}]

A364503 Sprague-Grundy values for Heat-Charge Toggle on paths from A364489 where paths with an even number of vertices are odious, or paths with an odd number of vertices are evil.

Original entry on oeis.org

0, 2, 1, 3, 2, 2, 3, 4, 5, 3, 7, 3, 4, 4, 6, 5, 5, 6, 4, 7, 6, 3, 10, 4, 16, 18, 16, 32, 32, 32, 32, 32, 36, 32, 36, 32, 33, 36, 32, 33, 32, 32, 32, 35, 33, 34, 34, 36, 35, 33, 33, 32, 36, 32, 33, 32, 36, 34, 34, 37, 37, 34

Offset: 1

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

See A364489 for the rules of Heat-Charge Toggle.

			a(2) = 2, which is odious. So the second term of A364489 is an even number.

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

Nathan Hurtig, C++ program to compute values

Cf. A000069, A001969, A363934. Sprague-Grundy values of A364489.

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.

Original entry on oeis.org

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

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

nonn

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.

			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).

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

Nathan Hurtig, C++ program to compute values

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

A363934 Table read by ascending antidiagonals. T(n,k) is the Sprague-Grundy value for the Heat Toggle game played on an n X k grid where each vertex has initial weight 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0, 1, 0, 2, 0, 3, 1, 1, 3, 0, 3, 1, 3, 0, 3, 1, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 2, 3, 0, 0, 0, 0, 0, 0, 3, 2, 2, 3, 1, 1

Offset: 1

Jean-Pierre Appel, Patrick G. Cesarz, Djeneba Diop, Eugene Fiorini, Nathan Hurtig, Andrew Woldar, Jun 28 2023

Heat Toggle is an impartial two-player game played on a simple graph, where each vertex is assigned a weight of -1 or 1.
A Heat Toggle move consists of selecting a vertex of weight 1 and switching its weight to -1 as well as switching the weight of each of its neighbors, changing 1 to -1 and -1 to 1. We additionally only allow moves that strictly decrease the sum of all weights.
The first row T(1,k) coincides with octal game 0.1337, see A071427.
The second row T(2,k) coincides with the octal game 0.137 (Dawson's Chess), see A002187.

			The data is organized in a table beginning with row n = 1 and column k = 1. The data is read by ascending antidiagonals. T(2,3)=2.
The table T(n,k) begins:
[n/k]  1   2   3   4   5   6  ...
---------------------------------
[1]    1,  1,  1,  2,  2,  0, ...
[2]    1,  1,  2,  0,  3,  1, ...
[3]    1,  2,  1,  1,  3,  0, ...
[4]    2,  0,  1,  0,  1,  0, ...
[5]    2,  3,  3,  1,  2,  0, ...
[6]    0,  1,  0,  0,  0,  ...

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

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. A361517, A071427, A002187.

SG_value_hash = {}
def MEX(S):
    i = 0
    while True:
        if i not in S:
            return i
        i += 1
def SG_value(G):
    global SG_value_hash
    SG_value_hash = {}
    ons = set(G.vertices())
    offs = set()
    return SG_value_helper(G, ons, offs)
def SG_value_helper(G, ons, offs):
    ons_orig = ons.copy()
    offs_orig = offs.copy()
    child_SG_values = set()
    for v in ons_orig:
        vNeighborhood = set(G.neighbors(v))
        neighNowOff = ons_orig.intersection(vNeighborhood)
        neighNowOn = offs_orig.intersection(vNeighborhood)
        if len(neighNowOff) >= len(neighNowOn):
            ons.remove(v)
            offs.add(v)
            ons.update(neighNowOn)
            offs -= neighNowOn
            offs.update(neighNowOff)
            ons -= neighNowOff
            result = -1 # placeholder
            encoded_position = str(offs)
            if encoded_position in SG_value_hash:
                result = SG_value_hash[encoded_position]
            else:
                result = SG_value_helper(G, ons, offs)
            SG_value_hash[encoded_position] = result
            ons.add(v)
            offs.remove(v)
            ons -= neighNowOn
            offs.update(neighNowOn)
            offs -= neighNowOff
            ons.update(neighNowOff)
            child_SG_values.add(result)
    return MEX(child_SG_values)
for sum_of_both in range(2,11):
    antidiagonal = []
    for n in range(1, sum_of_both):
        G = graphs.Grid2dGraph(n, sum_of_both-n)
        antidiagonal.append(SG_value(G))
    print(antidiagonal)

cp's OEIS Frontend

User: Patrick G. Cesarz

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Python

A363894 Number of weakly connected components of a halved addsub configuration graph with respect to integers mod n over a path with two vertices.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A363893 Number of weakly connected components of an addsub configuration graph with respect to integers mod n over a path with two vertices.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A364503 Sprague-Grundy values for Heat-Charge Toggle on paths from A364489 where paths with an even number of vertices are odious, or paths with an odd number of vertices are evil.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A363934 Table read by ascending antidiagonals. T(n,k) is the Sprague-Grundy value for the Heat Toggle game played on an n X k grid where each vertex has initial weight 1.

Views

Author

Keywords

Comments

Examples

References

Links