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A360501 Number of edges added at n-th generation of hexagonal graph constructed in first quadrant (see Comments for precise definition).

%I A360501 #42 Feb 20 2023 20:25:07
%S A360501 0,1,1,2,4,5,6,7,8,10,10,12,12,15,14,17,16,20,18,22,20,25,22,27,24,30,
%T A360501 26,32,28,35,30,37,32,40,34,42,36,45,38,47,40,50,42,52,44,55,46,57,48,
%U A360501 60,50,62,52,65,54,67,56,70,58,72,60,75,62,77,64,80,66,82,68,85,70,87,72,90
%N A360501 Number of edges added at n-th generation of hexagonal graph constructed in first quadrant (see Comments for precise definition).
%C A360501 We build up a planar graph with hexagonal cells, based on the square grid. There are six kinds of edges.
%C A360501 A U edge is drawn from (x,y) to (x,y+1);
%C A360501 a U^{-1} edge is drawn from (x,y) to (x,y-1);
%C A360501 an L edge is drawn from (x,y) to (x-1,y+1);
%C A360501 an L^{-1} edge is drawn from (x,y) to (x+1,y-1);
%C A360501 an R edge is drawn from (x,y) to (x+1,y+1); and
%C A360501 an R^{-1} edge is drawn from (x,y) to (x-1,y-1).
%C A360501 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).
%C A360501 The graph is then extended using the following rules.
%C A360501 Every U is followed by a pair of lines, L and R;
%C A360501 every L is followed by a U;
%C A360501 every L^{-1} is followed by a pair U^{-1} and R; and
%C A360501 every R is followed by a pair U and L^{-1}.
%C A360501 Lines that fall outside the first quadrant are ignored, and lines that would coincide with existing lines are ignored.
%C A360501 Lines of type U^{-1} and R^{-1} do not need to be followed by anything.
%C A360501 The node numbers in the illustration indicate at which generation the node is reached. This is also the graph distance from the origin.
%C A360501 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.
%C A360501 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.
%C A360501 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.
%H A360501 N. J. A. Sloane, <a href="/A360501/a360501.pdf">Illustration of generations 0 to 16 of the construction</a>. The structure has quasi-period 4, that is, the structure essentially repeats itself every four generations, as suggested by the color scheme.
%F A360501 G.f.: x*(1+x+x^2+3*x^3+2*x^4+x^5+x^6-x^7)/((1-x^2)*(1-x^4)).
%Y A360501 Cf. A360512 (partial sums), A106466, A265428, A265429.
%Y A360501 Inspired by A182838.
%K A360501 nonn
%O A360501 0,4
%A A360501 _N. J. A. Sloane_, Feb 09 2023