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A353876
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Irregular table read by rows: for each internal vertex of a regular n-gon with all diagonals drawn remove all the edges connected directly to the vertex and then count the number of sides in the polygon that surrounds it; row n gives the number of resulting k-sided polygons, for k>=4, for all internal vertices.
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3
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0, 1, 0, 0, 5, 0, 6, 6, 0, 0, 0, 0, 0, 1, 0, 7, 0, 14, 0, 7, 7, 0, 8, 24, 8, 9, 0, 9, 18, 18, 0, 63, 0, 18, 0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 44, 33, 55, 143, 11, 22, 0, 11, 12, 24, 144, 24, 60, 0, 36, 0, 1, 0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13
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OFFSET
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3,5
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COMMENTS
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Numerous patterns are found in the values of the k-gons for different n. For example for n = 4*m + 2, with m>=1, there is one maximum sided k-gon with 2*n edges. For n = 4*m, with m>=3, there is one maximum sided k-gon with n edges. For odd n, where n>=11, there is n maximum sided k-gons with n+2 edges.
The 8-gon appears to be unique in that there is 9 maximum sided k-gons, k=8, which is not 1 or a multiple of 8.
Only a limit number of even-n n-gons have vertex-surrounding polygons with 4 edges, the minimum possible value. See A353991.
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LINKS
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Scott R. Shannon, Image of the 7-gon. In this and other images the vertex color is based on the surrounding polygon edge count shown in the key.
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FORMULA
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Sum of terms in row n = A007569(n) - n.
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EXAMPLE
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The 7-gon has seven internal vertices surrounded by polygons with 5 edges, fourteen internal vertices surrounded by polygons with 7 edges, seven internal vertices surrounded by polygons with 9 edges, and seven internal vertices surrounded by polygons with 10 edges, so row 7 is [0, 7, 0, 14, 0, 7, 7].
The table begins:
0;
1;
0, 0, 5;
0, 6, 6, 0, 0, 0, 0, 0, 1;
0, 7, 0, 14, 0, 7, 7;
0, 8, 24, 8, 9;
0, 9, 18, 18, 0, 63, 0, 18;
0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 11, 44, 33, 55, 143, 11, 22, 0, 11;
12, 24, 144, 24, 60, 0, 36, 0, 1;
0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13;
0, 14, 182, 196, 168, 126, 56, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \
0, 0, 1;
0, 15, 120, 90, 345, 525, 135, 105, 15, 0, 0, 0, 0, 15;
0, 32, 256, 240, 480, 224, 96, 16, 32, 0, 0, 0, 1;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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