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A338421 Place four points evenly spaced on a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.

%I A338421 #16 Oct 29 2020 19:24:31
%S A338421 1,4,21,16,57,44,93,36,149,132,217,176,301,268,385,208,489,452,605,
%T A338421 528,737,684,869,532,1021,964,1185,1072,1365,1292,1545,1112,1745,1668,
%U A338421 1957,1808,2185,2092,2413,1844,2661,2564,2921,2736,3197,3084,3473,2696,3769
%N A338421 Place four points evenly spaced on a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.
%C A338421 The rays are evenly spaced around each point. The first ray from each point goes opposite to the direction to the center of the circle. Should a ray hit another point it is terminated there.
%C A338421 To produce the illustrations below, all pairwise intersections between the rays are calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit.
%H A338421 Lars Blomberg, <a href="/A338421/b338421.txt">Table of n, a(n) for n = 1..642</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421.png">Illustration for n=3</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_1.png">Illustration for n=7</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_2.png">Illustration for n=8</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_3.png">Illustration for n=16</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_4.png">Illustration for n=22</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_5.png">Illustration for n=26</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_6.png">Illustration for n=27</a>
%H A338421 Lars Blomberg, <a href="/A338421/a338421_7.png">Illustration for n=38</a>
%F A338421 Conjectured for 3 <= n <= 642.
%F A338421 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F A338421 +=================================+
%F A338421 |      r |  m | a |   b |   c | d |
%F A338421 +---------------------------------+
%F A338421 |      2 |  4 | 3 |  -4 |   4 | 2 |
%F A338421 |      1 |  8 | 3 |   7 |  -8 | 2 |
%F A338421 |      3 |  8 | 3 |   7 |  -6 | 2 |
%F A338421 |      4 |  8 | 3 |  -8 |  16 | 2 |
%F A338421 |      5 |  8 | 3 |   7 |   4 | 2 |
%F A338421 |      7 |  8 | 3 |   7 | -10 | 2 |
%F A338421 |      0 | 48 | 3 | -31 | -32 | 2 |
%F A338421 |  8, 40 | 48 | 3 | -31 | 128 | 2 |
%F A338421 | 16, 32 | 48 | 3 | -31 | 144 | 2 |
%F A338421 |     24 | 48 | 3 | -31 |  80 | 2 |
%F A338421 +=================================+
%e A338421 For n=1 there are four rays that do not intersect, so a(1)=1.
%o A338421 (PARI)
%o A338421 a(n)={if(
%o A338421 n==1,1,
%o A338421 n==2,4,
%o A338421 n%4==2,(3*n^2 - 4*n + 4)/2,
%o A338421 n%8==1,(3*n^2 + 7*n - 8)/2,
%o A338421 n%8==3,(3*n^2 + 7*n - 6)/2,
%o A338421 n%8==4,(3*n^2 - 8*n + 16)/2,
%o A338421 n%8==5,(3*n^2 + 7*n + 4)/2,
%o A338421 n%8==7,(3*n^2 + 7*n - 10)/2,
%o A338421 n%48==0,(3*n^2 - 31*n - 32)/2,
%o A338421 n%48==8||n%48==40,(3*n^2 - 31*n + 128)/2,
%o A338421 n%48==16||n%48==32,(3*n^2 - 31*n + 144)/2,
%o A338421 n%48==24,(3*n^2 - 31*n + 80)/2,
%o A338421 -1);}
%o A338421 vector(642, n, a(n))
%Y A338421 Cf. A338122, A338422 (vertices), A338423 (edges).
%K A338421 nonn
%O A338421 1,2
%A A338421 _Lars Blomberg_, Oct 26 2020