cp's OEIS Frontend

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

%I A363894 #20 Sep 27 2023 15:53:02
%S A363894 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,
%T A363894 25,1,29,19,58,5,42,6,75,8,91,20,34,4,108,7,13,2,17,39,164,18,58,14,
%U A363894 83,3,47,32,16,33,66,25,167,1,365,29,18,19,56,58,19,5
%N A363894 Number of weakly connected components of a halved addsub configuration graph with respect to integers mod n over a path with two vertices.
%C A363894 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).
%C A363894 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.
%C A363894 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).
%D A363894 E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
%H A363894 E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Wong/wong31.pdf">Characterizing Winning Positions in the Impartial Two-player Pebbling Game on Complete Graphs</a>, Journal of Integer Sequences, 24(6) (2021).
%H A363894 E. Fiorini, M. Lind, and A. Woldar, <a href="https://doi.org/10.1007/s00373-021-02453-z">On Properties of Pebble Assignment Graphs</a>, Graphs and Combinatorics, 38(2) (2022), 45.
%H A363894 E. Fiorini, G. Johnston, M. Lind, A. Woldar, and T. W. H. Wong, <a href="https://doi.org/10.1007/s00373-022-02552-5">Cycles and Girth in Pebble Assignment Graphs</a>, Graphs and Combinatorics, 38(5) (2022), 154.
%t A363894 Upto=25;
%t A363894 Table[
%t A363894   VertexSet:={};
%t A363894   EdgeSet:={};
%t A363894   (* Compute configuration graph for integers mod n *)
%t A363894   Do[
%t A363894     Do[AppendTo[VertexSet,{i,j}];
%t A363894       AppendTo[EdgeSet,{i,j}\[DirectedEdge]{Mod[i-j,n],Mod[i+j,n]}],
%t A363894       {j,0,n-1}],
%t A363894     {i,0,n-1}];
%t A363894   (* Print n-th term *)
%t A363894   Length[WeaklyConnectedComponents[Graph[VertexSet,EdgeSet]]],
%t A363894   {n,2,Upto}]
%Y A363894 Cf. A340631, A346197, A346401, A363893.
%K A363894 nonn
%O A363894 2,2
%A A363894 _Patrick G. Cesarz_, _Eugene Fiorini_, _Charles Gong_, _Kyle A. Kelley_, _Philip Thomas_, and _Andrew Woldar_, Jul 23 2023