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A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

%I A351045 #29 Jan 31 2022 20:53:25
%S A351045 3,4,0,4,5,0,10,0,5,6,0,18,12,6,7,0,28,14,21,14,7,8,0,56,48,24,9,0,54,
%T A351045 54,72,72,18,0,9,10,0,80,160,120,20,11,0,88,154,198,198,55,0,0,0,11,
%U A351045 12,0,240,336,168,13,0,130,260,507,390,91,104,0,0,0,0,13,14,0,266,616,644,140,42
%N A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.
%C A351045 The number of facing edges for a given edge is the number of other edges in the one (for edges on the outside of the n-gon) or two polygons that the edge forms a part of. For example, for an edge shared between two adjoined triangles the number of facing edges is four, as it faces two edges in each of the two triangles it forms a part of.
%C A351045 All edges that are on the outside of the n-gon have two facing edges as any such edge belongs to only one (interior) triangle. Thus T(n,2) = n. For odd n the central created n-gon, see A342222, is surrounded by triangles, thus the edges that form this central n-gon have (n-1)+(3-1) = n+1 facing edges, thus T(n,n+1) >= n.
%C A351045 For all n-gons with even n, or odd n if the central n-gon is ignored, the maximum k for which row(n,k) > 0 is unknown, although it is clearly related to the maximum sided cell for all n-gons; see A349784.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045.txt">Table for n = 3..100</a>.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045.gif">Image of the edges for n = 6</a>.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045_1.gif">Image of the edges for n = 9</a>.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045_2.gif">Image of the edges for n = 15</a>.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045_3.gif">Image of the edges for n = 25</a>.
%H A351045 Scott R. Shannon, <a href="/A351045/a351045_4.gif">Image of the edges for n = 30</a>.
%F A351045 Sum of row n = A135565(n).
%F A351045 T(n,2) = n.
%F A351045 T(n,n+1) >= n for odd n.
%e A351045 A hexagon with all diagonals drawn has six edges (those on the outside of the hexagon) which form one side of a single triangle and thus face two edges, eighteen edges that adjoin two triangles and thus face four edges, twelve edges that adjoin a triangle and a quadrilateral and thus face five edges, and six edges that adjoin two quadrilaterals and thus face six edges. Thus the row for n = 6 is [6, 0, 18, 12, 6]. See the attached image.
%e A351045 The table begins:
%e A351045 3;
%e A351045 4, 0, 4;
%e A351045 5, 0, 10, 0, 5;
%e A351045 6, 0, 18, 12, 6;
%e A351045 7, 0, 28, 14, 21, 14, 7;
%e A351045 8, 0, 56, 48, 24;
%e A351045 9, 0, 54, 54, 72, 72, 18, 0, 9;
%e A351045 10, 0, 80, 160, 120, 20;
%e A351045 11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11;
%e A351045 12, 0, 240, 336, 168;
%e A351045 13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13;
%e A351045 14, 0, 266, 616, 644, 140, 42;
%e A351045 15, 0, 180, 600, 945, 630, 435, 0, 15, 0, 0, 0, 0, 0, 15;
%e A351045 16, 0, 448, 1056, 960, 576, 32;
%e A351045 17, 0, 238, 816, 1853, 1224, 425, 272, 34, 0, 0, 0, 0, 0, 0, 0, 17;
%e A351045 18, 0, 900, 1836, 1314, 108, 144;
%e A351045 19, 0, 304, 1520, 2717, 2128, 798, 304, 95, 0, 19, 0, 0, 0, 0, 0, 0, 0, 19;
%e A351045 20, 0, 1000, 2120, 3280, 1600, 100, 240;
%e A351045 21, 0, 378, 2352, 4494, 3276, 1365, 252, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21;
%e A351045 22, 0, 1056, 3828, 5258, 1716, 374, 396, 132;
%e A351045 .
%e A351045 .
%e A351045 See the linked file for the table n = 3..100.
%Y A351045 Cf. A135565, A342222, A349784, A344899, A342236, A342222, A342236.
%K A351045 nonn,tabf
%O A351045 3,1
%A A351045 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 30 2022