A274585 -id:A274585 - cp's OEIS frontend

A366485 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of edges in the resulting planar graph.

3, 9, 48, 237, 684, 1962, 3630, 7617, 12654, 21114, 31170, 50280, 66687, 99342, 132756, 174567, 222495, 302553, 367158, 479226, 579057, 705432, 846477, 1055679, 1217541, 1460205, 1715088, 2011161, 2289753, 2729301, 3044637, 3561606, 4037604, 4587153, 5175597, 5865729, 6432138, 7327737

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

See A366483 for further information. See A366483 and A366486 for images of the triangle.

Cf. A366483 (vertices), A366484 (interior vertices), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366483(n) + A366486(n) - 1 (Euler).

A367322 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 10, 5, 58, 37, 10, 178, 257, 121, 19, 558, 817, 1055, 301, 42, 1255, 2757, 3506, 1753, 708, 57, 2532, 4825, 10410, 6913, 5369, 1145, 135, 4786, 12293, 21111, 17713, 17417, 8417, 2395, 171, 7804, 19241, 43740, 38497, 47796, 29121, 16434, 3581, 341

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

See A367323 and the cross references for further images of the n-gons.

			The table begins:
3, 10, 58, 178, 558, 1255, 2532, 4786, 7804, 12292, 18966, 28540, 39117, 56107, ...
5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, ...
10, 121, 1055, 3506, 10410, 21111, 43740, 74526, 124490, 190291, 288190, ...
19, 301, 1753, 6913, 17713, 38497, 80473, 139927, 225595, 356329, 549967, ...
42, 708, 5369, 17417, 47796, 99261, 194278, 331955, 546805, 833946, 1245314, ...
57, 1145, 8417, 29121, 80345, 167105, 333297, 570969, 939113, 1441153, 2153937, ...
135, 2395, 16434, 53155, 141147, 293374, 565767, 966493, 1580940, 2411533, ...
171, 3581, 23651, 80191, 213041, 444251, 862711, 1481141, 2413721, 3701951, ...
341, 6062, 39248, 126061, 329131, 684223, 1307845, 2233815, 3639020, 5549952, ...
313, 7513, 47293, 167941, 450457, 931345, 1830625, 3132349, 5103589, 7825201, ...
728, 12845, 79859, 255711, 660140, 1372008, 2608476, 4454477, 7236853, ...
771, 16871, 103517, 343855, 881959, 1847525, 3504971, 6013953, 9739227, ...
1380, 24136, 145635, 465721, 1192710, 2478121, 4694040, 8014891, 12995535, ...
1393, 30305, 182785, 602337, 1533681, 3211873, 6067041, 10402769, 16824161, ...
2397, 41583, 245684, 783803, 1995341, 4145230, 7829333, 13366897, ...
1855, 46801, 291637, 967123, 2476873, 5166055, 9798175, ...
3895, 67090, 389690, 1241765, 3146533, 6535659, 12317567, ...
3861, 80921, 466141, 1522001, 3840181, 8027441, 15094521, ...
6006, 102817, 589029, 1875511, 4734786, 9833104, ...
5963, 122849, 694387, 2259005, 5671887, 11852105, ...
8878, 151180, 856589, 2725041, 6858600, 14242153, ...
7321, 169297, 979777, 3205921, 8068321, ...
12675, 214851, 1206050, 3834451, 9626475, ...
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Scott R. Shannon, Image for T(12,2).
Scott R. Shannon, Image for T(23,1).

Cf. A367323 (regions), A367324 (edges), A274585 (1st row), A331449 (2nd row), A330847 (3rd row), A330846 (4th row), A333113 (5th row), A333109 (6th row), A332428 (7th row), A332418 (8th row), A007569 (1st column).

T(n,k) = A367324(n,k) - A367323(n,k) + 1 (Euler).

cp's OEIS Frontend

A366485 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of edges in the resulting planar graph.

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