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Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.
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%I #37 Aug 29 2020 02:51:50

%S 0,4,10,0,5,18,12,28,14,21,0,7,56,48,54,54,54,36,0,0,9,180,160,30,88,

%T 154,88,88,0,0,0,0,11,240,336,130,260,299,182,0,104,0,0,0,0,13,266,

%U 616,266,180,600,375,300,90,0,0,0,0,0,0,0,15,448,1056,320,256

%N Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.

%C See A335614 for the number of 4-gon to k-gon contacts, with k>=4.

%C See A335646 for the number of 5-gon to k-gon contacts, with k>=5.

%C See A337330 for the number of 6-gon to k-gon contacts, with k>=6.

%C See A007678 for the number of regions and images of other n-gons.

%H Scott R. Shannon, <a href="/A333654/a333654.png">Image for n=4 with edge-count coloring</a>. This is the first n-gon with 3-gon to 3-gon contacts.

%H Scott R. Shannon, <a href="/A333654/a333654_1.png">Image for n=6 with edge-count coloring</a>. This is the first n-gon with 3-gon to 4-gon contacts.

%H Scott R. Shannon, <a href="/A333654/a333654_2.png">Image for n=7 with edge-count coloring</a>. This is the first n-gon with 3-gon to 5-gon contacts, other than n=5 which has an internal 5-gon surrounded by 3-gons by its construction.

%H Scott R. Shannon, <a href="/A333654/a333654_3.png">Image for n=9 with edge-count coloring</a>. This is the first n-gon with 3-gon to 6-gon contacts.

%H Scott R. Shannon, <a href="/A333654/a333654_4.png">Image for n=15 with edge-count coloring</a>. This is the first n-gon with 3-gon to 7-gon contacts, other than n=7 which has an internal 7-gon surrounded by 3-gons by its construction.

%H Scott R. Shannon, <a href="/A333654/a333654_8.png">Image for n=15 with random distance-based coloring</a>.

%H Scott R. Shannon, <a href="/A333654/a333654_6.png">Image for n=13 with edge-count coloring</a>. This is the first n-gon with 3-gon to 8-gon contacts.

%H Scott R. Shannon, <a href="/A333654/a333654_7.png">Image for n=29 with edge-count coloring</a>. This is the first n-gon with 3-gon to 10-gon and 3-gon to 11-gon contacts.

%F For odd n, with n > 3, row(n,n) = n as the n-gon contains a smaller n-gon at its center surrounded by 3-gons.

%e The table begins:

%e .

%e 0;

%e 4;

%e 10,0,5;

%e 18,12;

%e 28,14,21,0,7;

%e 56,48;

%e 54,54,54,36,0,0,9;

%e 180,160,30;

%e 88,154,88,88,0,0,0,0,11;

%e 240,336;

%e 130,260,299,182,0,104,0,0,0,0,13;

%e 266,616,266;

%e 180,600,375,300,90,0,0,0,0,0,0,0,15;

%e 448,1056,320,256;

%e 238,816,935,578,85,238,0,0,0,0,0,0,0,0,17;

%e 900,1836,324,108,126;

%e 304,1520,1463,798,304,228,0,0,0,0,0,0,0,0,0,0,19;

%e 1000,2120,1740,560,0,160;

%e 378,2352,2016,1554,84,0,0,0,0,0,0,0,0,0,0,0,0,0,21;

%e 1056,3828,2310,308,176,220;

%e 460,2852,4117,2530,644,138,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23;

%e 2352,5856,2376,288;

%e 550,4550,4800,3350,900,250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25;

%e 1950,6708,5772,1612,650,208;

%e 648,6372,6399,4968,432,810,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27;

%e 2380,10024,5712,3584,616,224;

%e 754,8642,8497,6148,783,928,0,870,232,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29;

%e 6330,14640,5340,2340;

%e 868,11532,11191,7378,806,1302,0,496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31;

%e 3712,16384,12000,6016,1248;

%e 990,14454,14619,12540,2079,1782,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,33;

%Y Cf. A335614 (4-gon contacts), A335646 (5-gon contacts), A337330 (6-gon contacts), A007678, A135565, A007569, A062361, A331450, A331451.

%K nonn,tabf

%O 3,2

%A _Scott R. Shannon_, Aug 23 2020