A360044 Table read by rows: T(n,k) is the number of k-gons, 3<=k<=4, in a Farey fan of order n.

0, 1, 4, 0, 6, 2, 10, 4, 14, 10, 22, 14, 30, 24, 42, 34, 54, 50, 74, 62, 94, 84, 118, 106, 142, 140, 178, 168, 214, 204, 258, 240, 302, 292, 358, 338, 414, 402, 478, 466, 542, 542, 626, 608, 710, 696, 802, 784, 894, 892, 1010, 988, 1126, 1102, 1254, 1216, 1382, 1358, 1526, 1492

Offset: 1

Author

Scott R. Shannon, N. J. A. Sloane and M. Douglas McIlroy , Jan 23 2023

Keywords

• nonn
• tabf

Comments

See the reference for the definition of a 'Farey fan', along with a proof that only 3-gons and 4-gons are created. See A360042 for further details and images of the graph.

Examples

			The table begins:
0, 1;
4, 0;
6, 2;
10, 4;
14, 10;
22, 14;
30, 24;
42, 34;
54, 50;
74, 62;
94, 84;
118, 106;
142, 140;
178, 168;
214, 204;
.
.
		

Links

• M. D. McIlroy, A Note on Discrete Representation of Lines, AT&T Technical Journal, 64 (1985), 481-490.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 10.

Crossrefs

Cf. A005598 (regions), A360042 (vertices), A360043 (edges), A005728, A174030, A359977, A359971, A359694.