A367276 -id:A367276 - cp's OEIS frontend

A367277 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of interior vertices in the resulting planar graph.

0, 1, 57, 329, 1317, 2861, 7417, 12801, 23329, 36549, 64625, 80085, 138261, 176829, 233705, 311949, 455205, 513205, 741889, 819509, 1046161, 1310673, 1692869, 1772001, 2315189, 2713893, 3165017, 3552641, 4538729, 4602865, 6015437, 6432553, 7421213, 8550349, 9439481, 10063849, 12635621

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

See A367276 for further information.

Scott R. Shannon, Image for n = 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 = 10.

Cf. A367276 (interior), A367278 (regions), A367279 (edges).

a(n) = A367279(n) - A367278(n) - 4*n - 3 (Euler).

A367278 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 8, 88, 444, 1544, 3584, 8356, 14996, 26572, 42144, 69988, 93264, 148364, 196120, 261536, 347544, 483056, 572832, 783252, 907472, 1141668, 1413512, 1775972, 1958684, 2466452, 2897004, 3384128, 3852068, 4733632, 5042324, 6263264, 6869784, 7878988, 9014840, 10035784, 10909592, 13109608

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

See A367276 for further information.

Scott R. Shannon, Image for n = 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 = 9.
Scott R. Shannon, Image for n = 10.

Cf. A367276 (vertices), A367277 (interior vertices), A367279 (edges).

a(n) = A367279(n) - A367276(n) + 1 (Euler).

A367279 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

4, 16, 156, 788, 2880, 6468, 15800, 27828, 49936, 78732, 134656, 173396, 286676, 373004, 495300, 659556, 938328, 1086108, 1525216, 1727060, 2187912, 2724272, 3468932, 3730780, 4781740, 5611000, 6549252, 7404820, 9272476, 9645308, 12278824, 13302464, 15300332, 17565324, 19475404, 20973584

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

See A367276 for further information. See A367276 and A367278 for images of the square.

Cf. A367276 (vertices), A367277 (interior vertices), A367278 (regions).

a(n) = A367276(n) + A367278(n) - 1 (Euler).

A367302 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of these n*k 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, 6, 4, 22, 9, 5, 108, 69, 15, 6, 300, 345, 215, 25, 7, 919, 1337, 1285, 397, 49, 8, 1626, 2885, 4435, 2461, 1008, 65, 9, 3558, 7445, 11310, 7873, 5817, 1601, 144, 10, 5824, 12833, 24490, 21271, 19677, 9225, 3069, 181, 11, 9843, 23365, 46610, 46183, 49973, 33201, 17316, 4401, 352, 12

Offset: 3

Scott R. Shannon, Nov 13 2023

See A366483 and A367276 for other images of the n-gons.

			The table begins:
3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, ...
4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, ...
5, 15, 215, 1285, 4435, 11310, 24490, 46610, 81005, 131560, 202610, 298690, ...
6, 25, 397, 2461, 7873, 21271, 46183, 87475, 150445, 249985, 388885, 569839, ...
7, 49, 1008, 5817, 19677, 49973, 106169, 200564, 346682, 560672, 861329, ...
8, 65, 1601, 9225, 33201, 83361, 182705, 341433, 597169, 961761, 1490689, ...
9, 144, 3069, 17316, 57555, 145062, 306684, 576783, 994230, 1605357, 2462112, ...
10, 181, 4401, 25201, 87301, 218211, 469401, 877291, 1522231, 2452231, 3781541, ...
11, 352, 7326, 40568, 133793, 335192, 706387, 1324851, 2279794, 3676431, ...
12, 325, 7897, 53125, 182713, 456253, 990229, 1849549, 3207325, 5171497, ...
13, 741, 14963, 81757, 267995, 668811, 1406366, 2632708, 4524910, ...
14, 785, 19489, 107157, 360389, 893117, 1896665, 3536387, 6103889, ...
15, 1395, 27420, 148335, 484005, 1204395, 2528445, 4726770, 8116650, ...
.
.
.

Scott R. Shannon, Image for T(5,5).
Scott R. Shannon, Image for T(8,3).
Scott R. Shannon, Image for T(12,2).

Cf. A367303 (internal vertices), A367304 (regions), A367305 (edges), A366483 (first row), A367276 (second row).

a(n,k) = A367305(n,k) - A367304(n,k) + 1 (Euler).

cp's OEIS Frontend

A367277 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of interior vertices in the resulting planar graph.

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A367278 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of regions in the resulting planar graph.

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A367279 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of edges in the resulting planar graph.

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A367302 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of these n*k points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of vertices in the resulting planar graph.

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