|
%I #22 Jan 23 2024 10:58:31
%S 1,4,24,104,316,712,1588,2816,4940,7672,12444,16840,25968,34088,46260,
%T 61048,82792,98984,133032,156072,196236,239048,298292,334032,417072,
%U 483856,570200,649816,786412,850000,1037628,1145424,1311536,1485880,1677660,1828360,2158192,2357376,2623604,2852688
%N Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.
%H Scott R. Shannon, <a href="/A355798/a355798.jpg">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_1.jpg">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_2.jpg">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_3.jpg">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_4.jpg">Image for n = 6</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_5.jpg">Image for n = 7</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_6.jpg">Image for n = 8</a>.
%H Scott R. Shannon, <a href="/A355798/a355798_7.jpg">Image for n = 16</a>.
%H Scott R. Shannon, <a href="/A355798/a355798.png">Alternative version for n = 15</a>. The lines joining the vertices that are exactly opposite are omitted, forming a more carpet-like image. The number of regions formed, 42224, is less than the sequence version, although more 5,6,7 and 8-gon regions are present.
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, pp. 20-21.
%F a(n) = A355800(n) - A355799(n) + 1 by Euler's formula.
%Y Cf. A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.
%K nonn
%O 1,2
%A _Scott R. Shannon_, Jul 17 2022
|
