A358948
Number of regions formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).
Original entry on oeis.org
1, 12, 228, 1464, 12516, 29022, 153564, 364650, 996672, 1750326, 5274156, 7761498
Offset: 1
- Scott R. Shannon, Image for n = 2.
- 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.
- N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
- Wikipedia, Farey sequence.
A358951
Irregular table read by rows: T(n,k) = number of k-gons, k >= 3, formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,m)/A006843(n,m), m = 1..A005728(n).
Original entry on oeis.org
1, 12, 180, 42, 6, 810, 576, 72, 6, 6786, 4932, 744, 48, 6, 13662, 12522, 2568, 258, 12, 72582, 64932, 14376, 1632, 36, 6, 164484, 155088, 38688, 5958, 414, 18, 439524, 422370, 114804, 18462, 1392, 120, 750108, 749928, 211518, 35226, 3336, 204, 6, 2265462, 2240994, 647184, 109602, 10230, 666, 18
Offset: 1
The table begins:
1;
12;
180, 42, 6;
810, 576, 72, 6;
6786, 4932, 744, 48, 6;
13662, 12522, 2568, 258, 12;
72582, 64932, 14376, 1632, 36, 6;
164484, 155088, 38688, 5958, 414, 18;
439524, 422370, 114804, 18462, 1392, 120;
750108, 749928, 211518, 35226, 3336, 204, 6;
2265462, 2240994, 647184, 109602, 10230, 666, 18;
3263436, 3312270, 990072, 176172, 18294, 1188, 66;
.
.
A358950
Number of edges formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).
Original entry on oeis.org
3, 21, 375, 2574, 22083, 52791, 279750, 673050, 1851816, 3272058, 9865560, 14592537
Offset: 1
A359654
Number of vertices formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.
Original entry on oeis.org
4, 9, 77, 593, 6749, 15569, 93281, 222933, 623409, 1087393, 3453289, 5011009, 13271517
Offset: 1
A359968
Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.
Original entry on oeis.org
3, 6, 37, 195, 1467, 3408, 17113, 40435, 109638, 191718, 572939, 842487, 2139708, 3231583, 5261013
Offset: 1
A359974
Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.
Original entry on oeis.org
3, 6, 26, 93, 424, 876, 2785, 5542, 11575, 18761, 40249, 57399, 109376, 155965, 227884, 322377, 532454, 676282, 1056010, 1334975, 1767798, 2240664, 3252047, 3882192, 5226897
Offset: 1
- McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.
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