A346401 -id:A346401 - cp's OEIS frontend

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

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

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.

			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.

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

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.

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

CGSuite
```
# See Levandosky link
```

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.

Original entry on oeis.org

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.
```

A347637 Table read by ascending antidiagonals. T(n, k) is the minimum number of pebbles such that any assignment of those pebbles on a complete graph with n vertices is a next-player winning game in the two-player impartial (k+1, k) pebbling game. T(n, k) for n >= 5 and k >= 1.

Original entry on oeis.org

7, 13, 15, 9, 21, 21, 15, 17, 35, 27, 11, 25, 25, 37, 33, 17, 21, 41, 33, 59, 39, 13, 29, 31, 45, 41, 53

Offset: 5

Kayla Barker, Mia DeStefano, Eugene Fiorini, Michael Gohn, Joe Miller, Jacob Roeder, Wing Hong Tony Wong, Sep 09 2021

A (k+1, k) pebbling move involves removing k + 1 pebbles from a vertex in a simple graph and placing k pebbles on an adjacent vertex.
A two-player impartial (k+1, k) pebbling game involves two players alternating (k+1, k) pebbling moves. The first player unable to make a move loses.
T(3, k) = A016921(k) for k >= 0. The proof will appear in a paper that is currently in preparation.
It is conjectured that T(4, k) for odd k>=3 is infinite, so we start with n = 5.
T(5, k) = A346197(k) for k >= 1.
T(n, 1) = A340631(n) for n >= 3.
T(n, 2) = A346401(n) for n >= 3.

			The data is organized in a table beginning with row n = 5 and column k = 1. The data is read by ascending antidiagonals. The formula binomial(n + k - 5, 2) + k converts the indices from table form to sequence form.
The table T(n, k) begins:
  [n/k]  1   2   3   4   5   6  ...
  ---------------------------------
  [ 5]   7, 15, 21, 27, 33, 39, ...
  [ 6]  13, 21, 35, 37, 59, 53, ...
  [ 7]   9, 17, 25, 33, 41, 51, ...
  [ 8]  15, 25, 41, 45, 61, ...
  [ 9]  11, 21, 31, 41, 51, ...
  [10]  17, 29, 45, 53, 71, ...
  [11]  13, 25, 37, 49, 61, ...
  [12]  19, 33, 51, ...
  [13]  15, 29, 43, ...
  [14]  21, 37, ...
  [15]  17, 33, ...
  [16]  23, 41, ...

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

Kayla Barker, Mia DeStefano, Eugene Fiorini, Michael Gohn, Joe Miller, Jacob Roeder, and Tony W. H. Wong, Generalized Impartial Two-player Pebbling Games on K_3 and C_4, J. Int. Seq. (2024) Vol. 27, Issue 5, Art. No. 24.5.8. See p. 4.
Eugene Fiorini, Max Lind, Andrew Woldar, and Tony W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.6.4.

Cf. A340631, A346197, A346401.

(* m represents number of vertices in the complete graph. Each pebbling move removes k+1 pebbles from a vertex and adds k pebbles to an adjacent vertex. *)
Do[(* Given m and a, list all possible assignments with a pebbles. *)
alltuples[m_, a_] := IntegerPartitions[a + m, {m}] - 1;
(* Given an assignment, list all resultant assignments after one pebbling move; only works for m>=3. *)
pebblemoves[config_] :=
  Block[{m, temp}, m = Length[config];
   temp = Table[config, {i, m (m - 1)}] +
     Permutations[Join[{-(k + 1), k}, Table[0, {i, m - 2}]]];
   temp = Select[temp, Min[#] >= 0 &];
   temp = ReverseSort[DeleteDuplicates[ReverseSort /@ temp]]];
(* Given m and a, list all assignments that are P-games. *)
Plist = {};
plist[m_, a_] :=
  Block[{index, tuples},
   While[Length[Plist] < m, index = Length[Plist];
    AppendTo[Plist, {{Join[{1}, Table[0, {i, index}]]}}]];
   Do[AppendTo[Plist[[m]], {}]; tuples = alltuples[m, i];
    Do[If[
      Not[IntersectingQ[pebblemoves[tuples[[j]]],
        If[i > 2, Plist[[m, i - 1]], {}]]],
      AppendTo[Plist[[m, i]], tuples[[j]]]], {j, Length[tuples]}], {i,
      Length[Plist[[m]]] + 1, a}]; Plist[[m, a]]];
(* Given m, print out the minimum a such that there are no P-games with a pebbles *)
Do[a = 1; While[plist[m, a] != {}, a++];
  Print["k=", k, " m=", m, " a=", a], {m, 5, 10}], {k, 1, 6}]

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}]

A361315 a(n) is the minimum number of pebbles such that any assignment of those pebbles on a complete graph with n vertices is a next-player winning game in the two-player impartial (3;1,1) pebbling game.

Original entry on oeis.org

31, 26, 19, 17, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41

Offset: 4

Gabrielle Demchak, Eugene Fiorini, Michael J. Herrera, Samuel Murray, Rhaldni Sayaman, Brittany Shelton and Wing Hong Tony Wong, Mar 14 2023

A (3;1,1) move in an impartial two-player pebbling game consists of removing three pebbles from a vertex and adding a pebble to each of two distinct adjacent vertices. The winning player is the one who makes the final allowable move. We start at n = 4 because we have shown that a(3) does not exist while a(2) is clearly undefined.

			For n = 4, a(4) = 31 is the least number of pebbles for which every game is a next-player winning game regardless of assignment.

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

Eugene Fiorini, Max Lind, Andrew Woldar, and Tony W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.6.4.

Cf. A340631, A346197, A346401, A347637.

(*Given n and m, list all possible assignments.*)alltuples[n_, m_] := IntegerPartitions[m + n, {n}] - 1;
(*Given an assignment, list all resultant assignments after one (3;1,1)-pebbling move; only work for n>=3.*)
pebblemoves[config_] :=  Block[{n, temp}, n = Length[config];   temp = Table[config, {i, n (n - 1) (n - 2)/2}] +     Permutations[Join[{-3, 1, 1}, Table[0, {i, n - 3}]]];   temp = Select[temp, Min[#] >= 0 &];   temp = ReverseSort[DeleteDuplicates[ReverseSort /@ temp]]];
(*Given n and m, list all assignments that are P-games.*)
Plist = {};plist[n_, m_] :=  Block[{index, tuples},   While[Length[Plist] < n, index = Length[Plist];    AppendTo[Plist, {{Join[{1}, Table[0, {i, index}]]}}]];   Do[AppendTo[Plist[[n]], {}]; tuples = alltuples[n, i];    Do[If[Not[       IntersectingQ[pebblemoves[tuples[[j]]],        Plist[[n, i - 1]]]],      AppendTo[Plist[[n, i]], tuples[[j]]]], {j, Length[tuples]}], {i,      Length[Plist[[n]]] + 1, m}]; Plist[[n, m]]];
(*Given n, print out the minimum m such that there are no P-games with m pebbles*)Do[m = 1; While[plist[n, m] != {}, m++];
 Print["n=", n, " m=", m], {n, 4, 20}]

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}]

cp's OEIS Frontend

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.

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CGSuite

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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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.

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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.

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Python

A347637 Table read by ascending antidiagonals. T(n, k) is the minimum number of pebbles such that any assignment of those pebbles on a complete graph with n vertices is a next-player winning game in the two-player impartial (k+1, k) pebbling game. T(n, k) for n >= 5 and k >= 1.

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Mathematica

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

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Mathematica

A361315 a(n) is the minimum number of pebbles such that any assignment of those pebbles on a complete graph with n vertices is a next-player winning game in the two-player impartial (3;1,1) pebbling game.

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Mathematica

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

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