A368816 -id:A368816 - cp's OEIS frontend

A368813 Number of regions in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

0, 1, 13, 57, 171, 361, 813, 1489, 2215, 4081, 6249, 8329, 13027, 17977, 20221, 32033, 41583, 50545, 66881, 83161, 93871, 124521, 150237, 173233, 213351, 251473, 285445, 342889, 397011, 432121, 524149, 598081, 663103, 769217, 867441, 960121, 1091723, 1218889, 1333489, 1506321, 1667799

Offset: 1

Scott R. Shannon, Jan 06 2024

nonn

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.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 15.

Cf. A368814 (vertices), A368815 (edges), A368816 (k-gons), A368755, A007678.

a(n) = A368815(n) - A368814(n) + 1 by Euler's formula.

A368814 Number of vertices in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

Original entry on oeis.org

2, 4, 12, 48, 150, 288, 728, 1344, 1782, 3780, 5852, 7224, 12350, 17108, 16620, 30720, 40018, 46728, 64676, 80560, 84462, 121044, 146280, 163728, 208250, 245700, 271836, 335664, 389006, 404400, 514352, 587264, 638022, 756228, 853300, 933480, 1074998, 1200724, 1295112, 1485120, 1645002

Offset: 1

Scott R. Shannon, Jan 06 2024

nonn

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.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 15. Note that the maximum number of chord crossings on a single vertex is six for this 30-gon, which is one less than the maximum theoretical value of seven for the regular n-gon with all diagonals drawn; see A007569.

Cf. A368813 (regions), A368815 (edges), A368816 (k-gons), A368756, A007569.

a(n) = A368815(n) - A368813(n) + 1 by Euler's formula.

A368815 Number of edges in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

Original entry on oeis.org

0, 4, 24, 104, 320, 648, 1540, 2832, 3996, 7860, 12100, 15552, 25376, 35084, 36840, 62752, 81600, 97272, 131556, 163720, 178332, 245564, 296516, 336960, 421600, 497172, 557280, 678552, 786016, 836520, 1038500, 1185344, 1301124, 1525444, 1720740, 1893600, 2166720, 2419612, 2628600, 2991440

Offset: 1

Scott R. Shannon, Jan 06 2024

nonn

See A368813 and A368814 for images of the 2n-gons.

Cf. A368813 (regions), A368814 (vertices), A368816 (k-gons), A368757, A135565.

a(n) = A368813(n) + A368814(n) - 1 by Euler's formula.

cp's OEIS Frontend

A368813 Number of regions in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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A368814 Number of vertices in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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A368815 Number of edges in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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