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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: M. Douglas McIlroy

M. Douglas McIlroy's wiki page.

M. Douglas McIlroy has authored 3 sequences.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

Crossrefs

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

A360043 Number of edges in a Farey fan of order n.

Original entry on oeis.org

4, 9, 18, 30, 52, 74, 112, 154, 210, 268, 352, 436, 552, 668, 802, 948, 1134, 1316, 1546, 1778, 2038, 2306, 2630, 2952, 3326, 3704, 4124, 4556, 5060, 5552, 6126, 6710, 7338, 7978, 8674, 9376, 10174, 10972, 11824, 12692, 13664, 14620, 15690, 16768, 17898, 19048, 20314, 21574, 22944, 24312
Offset: 1

Views

Author

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

Keywords

Comments

See the reference for the definition of a 'Farey fan'. See A360042 and A360044 for further details and images of the graph.

References

  • McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

Crossrefs

Cf. A005598 (regions), A360042 (vertices), A360044 (k-gons), A005728, A174030, A359976, A359970, A359693.

A360042 Number of vertices in a Farey fan of order n.

Original entry on oeis.org

4, 6, 11, 17, 29, 39, 59, 79, 107, 133, 175, 213, 271, 323, 385, 451, 541, 621, 731, 835, 955, 1073, 1225, 1367, 1541, 1707, 1897, 2087, 2321, 2535, 2801, 3061, 3345, 3625, 3937, 4243, 4609, 4957, 5335, 5713, 6155, 6569, 7055, 7529, 8031, 8531, 9101, 9649, 10265, 10859
Offset: 1

Views

Author

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

Keywords

Comments

See the reference for the definition of a 'Farey fan'.
The number of vertices along each edge is A005728(n), while the number of regions is conjectured to equal A005598(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i). The regions count the number of distinct approximate representations of straight lines y = mx + b that can be drawn on an x-y integer raster, where x, y, and b are restricted to [0,n) and 0 <= m <=1.
It is also worth noting that for 3 <= n <= 10 this sequence equals 2*A005728(n) + A174030(n-2), where A174030(n) = Sum_{i=1..n} (i where phi(i)|i). That is, the number of internal vertices of the Farey fan equals A174030(n) in this range. This may suggest a possible attack on finding a formula for the present sequence.

Links

Crossrefs

Cf. A005598 (regions), A360043 (edges), A360044 (k-gons), A005728, A174030, A359974, A359968, A359690.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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