A363894 - OEIS

A363894

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

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

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OFFSET

2,2

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=2..72.

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.

MATHEMATICA

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 *)