A333654 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.

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, 154, 88, 88, 0, 0, 0, 0, 11, 240, 336, 130, 260, 299, 182, 0, 104, 0, 0, 0, 0, 13, 266, 616, 266, 180, 600, 375, 300, 90, 0, 0, 0, 0, 0, 0, 0, 15, 448, 1056, 320, 256

Offset: 3

Author

Scott R. Shannon, Aug 23 2020

Keywords

• nonn
• tabf

Comments

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

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

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

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

Examples

			The table begins:
.
0;
4;
10,0,5;
18,12;
28,14,21,0,7;
56,48;
54,54,54,36,0,0,9;
180,160,30;
88,154,88,88,0,0,0,0,11;
240,336;
130,260,299,182,0,104,0,0,0,0,13;
266,616,266;
180,600,375,300,90,0,0,0,0,0,0,0,15;
448,1056,320,256;
238,816,935,578,85,238,0,0,0,0,0,0,0,0,17;
900,1836,324,108,126;
304,1520,1463,798,304,228,0,0,0,0,0,0,0,0,0,0,19;
1000,2120,1740,560,0,160;
378,2352,2016,1554,84,0,0,0,0,0,0,0,0,0,0,0,0,0,21;
1056,3828,2310,308,176,220;
460,2852,4117,2530,644,138,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23;
2352,5856,2376,288;
550,4550,4800,3350,900,250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25;
1950,6708,5772,1612,650,208;
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;
2380,10024,5712,3584,616,224;
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;
6330,14640,5340,2340;
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;
3712,16384,12000,6016,1248;
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;
		

Links

• Scott R. Shannon, Image for n=4 with edge-count coloring. This is the first n-gon with 3-gon to 3-gon contacts.
• Scott R. Shannon, Image for n=6 with edge-count coloring. This is the first n-gon with 3-gon to 4-gon contacts.
• Scott R. Shannon, Image for n=7 with edge-count coloring. 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.
• Scott R. Shannon, Image for n=9 with edge-count coloring. This is the first n-gon with 3-gon to 6-gon contacts.
• Scott R. Shannon, Image for n=15 with edge-count coloring. 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.
• Scott R. Shannon, Image for n=15 with random distance-based coloring.
• Scott R. Shannon, Image for n=13 with edge-count coloring. This is the first n-gon with 3-gon to 8-gon contacts.
• Scott R. Shannon, Image for n=29 with edge-count coloring. This is the first n-gon with 3-gon to 10-gon and 3-gon to 11-gon contacts.

Crossrefs

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

Formula

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.