A345025 - OEIS

%I #17 Sep 12 2021 10:00:26

%S 1,2,7,16,36,72,141,232,424,630,1035,1284,2172,2716,4081,4848,7056,

%T 7290,11439,12960,17620,19712,26037,26568,37176,40638,51571,55832,

%U 69804,64440,92505,98912,120352,128146,154071,156348,194436,205352,242269,254920,298440,290766,363867,380776,439516

%N Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

%C The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity.

%C See A344857 for further examples and images of the regions.

%H Scott R. Shannon, <a href="/A345025/a345025.gif">Image for n = 3</a>. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges.

%H Scott R. Shannon, <a href="/A345025/a345025_1.gif">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_2.gif">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_3.gif">Image for n = 6</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_4.gif">Image for n = 7</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_5.gif">Image for n = 8</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_6.gif">Image for n = 9</a>.

%H Scott R. Shannon, <a href="/A345025/a345025_7.gif">Image for n = 10</a>.

%F Formula for odd n: a(n) = (n^4 - 7*n^3 + 27*n^2 - 29*n + 8)/8 (see A344857).

%F For n >= 3, a(n) = A344857(n) +  A002378(n-1).

%e a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions.

%e a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total.

%e a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.

%Y Cf. A344857 (number of polygons), A344311 (number polygons outside the n-gon), A007678 (number polygons inside the n-gon), A002378 (number of open regions for (n-1)-gon), A146212 (number of vertices), A344866, A344938.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jun 06 2021