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A351129 Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges. 2

%I #15 Feb 03 2022 02:54:11

%S 1,0,1,0,1,8,1,0,1,132,66,56,46,144,171,576,305,620,652,616,852,1296,

%T 1376,1482,1891,1820,2379,4530,3163,3328,3532,4046,4656,4896,6661,

%U 6460,7411,7560,9595,11676,10923,13552,10936,13294,14806,17232,17935,17200,20452,20540,24964,27270

%N Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges.

%C See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.

%H Scott R. Shannon, <a href="/A351129/b351129.txt">Table of n, a(n) for n = 3..100</a>

%H Scott R. Shannon, <a href="/A351129/a351129.gif">Image for n = 5</a>. In this and other images the regions with edges with the same facing edge count are highlighted in the corresponding edge color.

%H Scott R. Shannon, <a href="/A351129/a351129_1.gif">Image for n = 8</a>.

%H Scott R. Shannon, <a href="/A351129/a351129_2.gif">Image for n = 12</a>.

%H Scott R. Shannon, <a href="/A351129/a351129_3.gif">Image for n = 15</a>.

%H Scott R. Shannon, <a href="/A351129/a351129_4.gif">Image for n = 18</a>.

%H Scott R. Shannon, <a href="/A351129/a351129_5.gif">Image for n = 24</a>.

%e a(5) = 1. A pentagon with all diagonals drawn contains a central pentagon which is surrounded by five other triangles and therefore all its edges have a facing edge count of 6. See the attached image.

%e a(8) = 8. An octagon with all diagonals drawn contains eight central triangles all of which are surrounded by three other triangles and therefore all their edges have a facing edge count of 4. See the attached image.

%e a(15) = 46. A 15-gon with all diagonals drawn contains one central 15-gon which is surrounded by triangles, thirty quadrilaterals which are surrounded by other quadrilaterals, and fifteen triangles which are surrounded by pentagons. This gives a total of forty-six regions whose edges all have the same facing edge count. See the attached image.

%Y Cf. A351045, A135565, A007678, A342222, A349784, A344899, A342236.

%K nonn

%O 3,6

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 02 2022

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