A366253 Table read by antidiagonals: Place k points in general position 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 number of regions in the resulting planar graph.

1, 13, 4, 82, 67, 11, 307, 406, 206, 24, 841, 1441, 1216, 489, 50, 1891, 3796, 4211, 2835, 995, 80, 3718, 8299, 10901, 9672, 5671, 1802, 154, 6637, 15982, 23536, 24780, 19139, 10196, 3052, 220, 11017, 28081, 44906, 53109, 48686, 34166, 17011, 4810, 375

Offset

3,2

Comments

"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.

Note that although the number of regions with a given number of edges in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices created from the edge-point chords remain simple.

Links

Table of n, a(n) for n=3..47.

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

Scott R. Shannon, Image for T(6,2).

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

Scott R. Shannon, Image for T(10,2).

Formula

T(n,k) = A367190(n,k) - A367183(n,k) + 1 by Euler's formula.

Conjectured:

T(3,k) = A367118(k) = (9/4)*k^4 + 3*k^3 + (15/4)*k^2 + 3*k + 1.

T(4,k) = A367121(k) = (17/2)*k^4 + 19*k^3 + (43/2)*k^2 + 14*k + 4.

T(5,k) = (45/2)*k^4 + 60*k^3 + 70*k^2 + (85/2)*k + 11.

T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (687/4)*k^2 + 102*k + 24.

T(7,k) = (371/4)*k^4 + 287*k^3 + (1421/4)*k^2 + 210*k + 50.

T(8,k) = 161*k^4 + 518*k^3 + 655*k^2 + 388*k + 80.

T(9,k) = 261*k^4 + 864*k^3 + (2223/2)*k^2 + (1323/2)*k + 154.

T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (7085/4)*k^2 + 1060*k + 220.

Example

The table begins:
1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417,...
4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497,...
11, 206, 1216, 4211, 10901, 23536, 44906, 78341, 127711, 197426, 292436,...
24, 489, 2835, 9672, 24780, 53109, 100779, 175080, 284472, 438585, 648219,...
50, 995, 5671, 19139, 48686, 103825, 196295, 340061, 551314, 848471, 1252175,...
80, 1802, 10196, 34166, 86480, 183770, 346532, 599126, 969776, 1490570,...
154, 3052, 17011, 56611, 142696, 302374, 569017, 982261, 1588006, 2438416,...
220, 4810, 26705, 88495, 222400, 470270, 883585, 1523455, 2460620, 3775450,...
375, 7305, 40096, 132243, 331431, 699535, 1312620, 2260941, 3648943, 5595261,...
444, 10509, 57810, 190263, 475980, 1003269, 1880634, 3236775, 5220588, 8001165,...
781, 14938, 81082, 265747, 663391, 1396396, 2615068, 4497637, 7250257,...
952, 20335, 110439, 361354, 900844, 1894347, 3544975, 6093514, 9818424,...
1456, 27391, 147421, 480931, 1197076, 2514781, 4702741, 8079421, 13013056,...
1696, 35716, 192552, 627484, 1560352, 3275556, 6122056, 10513372,...
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Crossrefs

Cf. A367118 (first row), A367121 (second row), A007678 (first column), A367183 (vertices), A367190 (edges).

Sequence in context: A056139 A106293 A046734 * A226376 A222165 A277125

Adjacent sequences: A366250 A366251 A366252 * A366254 A366255 A366256

Keyword

nonn,tabl

Author

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

Status

approved