|
%I #21 Oct 11 2023 14:42:01
%S 1,7,25,55,97,151,217,295,385,475,601,715,865,1015,1159,1351,1537,
%T 1735,1945,2131,2401,2647,2905,3115,3457,3751,4057,4357,4705,5005,
%U 5401,5767,6133,6535,6925,7303,7777,8215,8653,9025,9601,10051,10585,11071,11587,12151,12697,13171,13825,14395,14989
%N Number of regions in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.
%C Unlike similar dissections of the triangle and square, see A356984 and A357058, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .
%H Scott R. Shannon, <a href="/A357196/a357196.jpg">Image for n = 1</a>.
%H Scott R. Shannon, <a href="/A357196/a357196_1.jpg">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A357196/a357196_2.jpg">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A357196/a357196_3.jpg">Image for n = 9</a>. This is the first term that forms hexagons with non-simple intersections.
%H Scott R. Shannon, <a href="/A357196/a357196_4.jpg">Image for n = 50</a>.
%H Scott R. Shannon, <a href="/A357196/a357196_5.jpg">Image for n = 150</a>.
%F a(n) = A357198(n) - A357197(n) + 1 by Euler's formula.
%F Conjecture: a(n) = 6*n^2 + 1 for hexagons that only contain simple intersections when cut by n internal hexagons.
%Y Cf. A357197 (vertices), A357198 (edges), A331931, A356984 (triangle), A357058 (square).
%Y Cf. A227776 (6*n^2 + 1).
%K nonn
%O 0,2
%A _Scott R. Shannon_, Sep 17 2022
|
