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We build up a planar graph with hexagonal cells, based on the square grid. There are six kinds of edges.

A U edge is drawn from (x,y) to (x,y+1);

a U^{-1} edge is drawn from (x,y) to (x,y-1);

an L edge is drawn from (x,y) to (x-1,y+1);

an L^{-1} edge is drawn from (x,y) to (x+1,y-1);

an R edge is drawn from (x,y) to (x+1,y+1); and

an R^{-1} edge is drawn from (x,y) to (x-1,y-1).

The construction starts in generation 0 with a single node at the origin (see illustration). At generation 1 we draw a U line from the origin to (0,1).

The graph is then extended using the following rules.

Every U is followed by a pair of lines, L and R;

every L is followed by a U;

every L^{-1} is followed by a pair U^{-1} and R; and

every R is followed by a pair U and L^{-1}.

Lines that fall outside the first quadrant are ignored, and lines that would coincide with existing lines are ignored.

Lines of type U^{-1} and R^{-1} do not need to be followed by anything.

The node numbers in the illustration indicate at which generation the node is reached. This is also the graph distance from the origin.

The number of nodes that are added at the n-th generation, for n >= 0, is given by 1, 1, 1, 2, 4, 4, 5, 5, 7, 7, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 16, 16, 17, 17, 19, 19, ..., with G.f. = (-x^7+x^6+x^4+x^3+1)/((1-x)*(1-x^4)). This is essentially A265428.

The total number of nodes after the n-th generation, for n >= 0, is 1, 2, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, ... This is essentially A265429.

The number of hexagons that are added at the n-th generation, for n >= 0, is given by 0, 0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 4, 8, 4, 9, 5, 10, 5, 11, 6, 12, 6, 13, ..., with G.f. = x^5*(1+x+x^2)/((1-x^2)*(1-x^4)). This is essentially A106466.