A346197 a(n) is the minimum number of pebbles such that any assignment of those pebbles on K_5 is a next-player winning game in the two-player impartial (n+1,n) pebbling game.

7, 15, 21, 27, 33, 39, 47, 53, 59, 67, 73, 79, 87, 93, 99, 107, 113, 119, 127, 133, 139

Offset: 1

Author

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

Keywords

• nonn
• more

Comments

For n>0, an (n+1,n) pebbling move involves removing n+1 pebbles from a vertex in a simple graph and placing n pebbles on an adjacent vertex.

A two-player impartial (n+1,n) pebbling game involves two players alternating (n+1,n) pebbling moves. The first player unable to make a move loses.

Examples

			For n=1, a(1)=7 is the least number of pebbles for which every (2,1) game on K_5 is a next-player winning game regardless of assignment.
		

References

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

Links

• 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, Journal of Integer Sequences, (2021) Vol. 24, Issue 6, Art. No. 21.6.4.

Crossrefs

Cf. A340631, A084964.

Programs

Mathematica

Do[remove = k + 1; add = k;
(*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 pebbling move; only work for n>=3.*)
pebblemoves[config_] :=  Block[{n, temp},
    n = Length[config];
    temp = Table[config, {i, n (n - 1)}] +
        Permutations[Join[{-remove, add}, Table[0, {i, n - 2}]]];
    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]]],
                If[i > (remove - add), Plist[[n, i - (remove - add)]], {}]]],
            AppendTo[Plist[[n, i]], tuples[[j]]]], {j, Length[tuples]}],
    {i, Length[Plist[[n]]] + 1, m}]; Plist[[n, m]]];
Do[m = 1; While[plist[n, m] != {}, m++]; Print[" k=", k, " m=", m], {n, 5, 5}],
{k, 1, 21}]