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.
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
Examples
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, ...
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, J. Int. Seq., Vol. 24 (2021), Article 21.6.4.
Programs
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Mathematica
(* 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}]
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