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A363893
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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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1
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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
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OFFSET
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2,2
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COMMENTS
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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.
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REFERENCES
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E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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