cp's OEIS Frontend

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.

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A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

Original entry on oeis.org

0, 0, 24, 228, 904, 2788, 5880, 13008, 22120, 37976, 58584, 95472, 125016, 195816, 255064, 337916, 444760, 611760, 719800, 978388, 1127088, 1411756, 1736776, 2174584, 2389552, 2995336, 3504768
Offset: 0

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Keywords

Comments

For n = 0 to n = 11 the number of polygons formed outside the n x n square is less than the number formed inside the square. This changes for n >= 12 when the number formed outside becomes greater. The ratio of the number of polygons outside to the number inside for n = 26 is about 1.07 . If this ratio is unbounded or approaches some finite value as n -> infinity is not known.
See A345459 for images of the polygons.

Crossrefs

Cf. A345459 (number inside and outside with square), A255011 (number inside the square), A344993, A344857, A092098, A007678.

Formula

a(n) = A345459(n) - A255011(n).

A367323 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 regions in the resulting planar graph.

Original entry on oeis.org

1, 12, 4, 75, 56, 11, 252, 340, 170, 24, 715, 1120, 1161, 408, 50, 1572, 3264, 3900, 2268, 868, 80, 3109, 6264, 10741, 8208, 5594, 1488, 154, 5676, 13968, 22380, 20832, 18396, 9312, 2754, 220, 9291, 22904, 44491, 44640, 48462, 31552, 16858, 4220, 375
Offset: 3

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Author

Keywords

Comments

See A367322 and the cross references for further images of the n-gons.

Examples

			The table begins:
1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, ...
4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, ...
11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, ...
24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, ...
50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, ...
80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, ...
154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, ...
220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, ...
375, 6732, 39887, 129492, 330903, 692648, 1311443, 2248840, 3645885, 5574756, ...
444, 9000, 52056, 178200, 462504, 963576, 1854432, 3180816, 5157612, 7906080, ...
781, 13962, 80783, 261222, 662663, 1385332, 2613521, 4478188, 7246331, ...
952, 18676, 107142, 352828, 891870, 1870876, 3525494, 6053768, 9778370, ...
1456, 25860, 146956, 474000, 1196116, 2498010, 4700776, 8050080, 13008106, ...
1696, 33152, 188000, 615328, 1547792, 3244000, 6095600, 10458560, 16876160, ...
2500, 44098, 247334, 795634, 1999762, 4173296, 7838004, 13416740, ...
2466, 52236, 302148, 991800, 2502000, 5229396, 9846234, ...
4029, 70604, 391781, 1258028, 3152101, 6574000, 12328417, ...
4500, 86240, 475800, 1546280, 3865240, 8085040, 15143880, ...
6175, 107562, 591655, 1897182, 4741633, 9883986, ...
6820, 129448, 706288, 2288880, 5703698, 11924132, ...
9086, 157412, 859718, 2753192, 6866858, 14307932, ...
9024, 181152, 1001640, 3254160, 8117304, ...
12926, 222850, 1209776, 3870250, 9636276, ...
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Crossrefs

Cf. A367322 (vertices), A367324 (edges), A092867 (1st row), A255011 (2nd row), A331929 (3rd row), A331931 (4th row), A329713 (5th row), A333075 (6th row), A332421 (7th row), A333139 (8th row), A007678 (1st column).

Formula

T(n,k) = A367324(n,k) - A367322(n,k) + 1 (Euler).
Previous Showing 31-32 of 32 results.