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A338124 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of edges thus created. See Comments for details.

%I A338124 #13 Oct 25 2020 14:17:10
%S A338124 3,9,24,36,63,60,120,114,150,171,237,138,321,321,375,378,522,456,651,
%T A338124 564,717,765,912,606,1068,1059,1158,1116,1413,1284,1614,1482,1716,
%U A338124 1791,2019,1470,2247,2229,2373,2322,2736,2544,3009,2796,3147,3249,3558,2802,3858
%N A338124 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of edges thus created. See Comments for details.
%C A338124 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 A338124 See A338122 for illustrations.
%H A338124 Lars Blomberg, <a href="/A338124/b338124.txt">Table of n, a(n) for n = 1..800</a>
%F A338124 a(n) = 4320-a(n-4)+a(n-12)+a(n-16)+a(n-60)+a(n-64)-a(n-72)-a(n-76), n>78. (conjectured)
%F A338124 From _Lars Blomberg_, Oct 25 2020: (Start)
%F A338124 Conjectured for 3 <= n <= 800.
%F A338124 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F A338124 +===========================================+
%F A338124 |               r |   m | a |   b |   c | d |
%F A338124 +-------------------------------------------+
%F A338124 |            1, 5 |  12 | 6 |  21 |  -3 | 4 |
%F A338124 |           2, 10 |  12 | 3 |   3 |  12 | 2 |
%F A338124 |               3 |  12 | 6 |   9 |  15 | 4 |
%F A338124 |               6 |  12 | 3 |  -6 |  48 | 2 |
%F A338124 |               7 |  12 | 6 |  21 |  39 | 4 |
%F A338124 |               9 |  12 | 6 |   9 |  33 | 4 |
%F A338124 |              11 |  12 | 6 |  21 |  -9 | 4 |
%F A338124 |           4, 20 |  24 | 3 |  -6 |  48 | 2 |
%F A338124 |           8, 16 |  24 | 3 |  -6 |  84 | 2 |
%F A338124 |               0 | 120 | 3 | -33 | -12 | 2 |
%F A338124 | 12, 36, 84, 108 | 120 | 3 | -33 | 240 | 2 |
%F A338124 |  24, 48, 72, 96 | 120 | 3 | -33 | 276 | 2 |
%F A338124 |              60 | 120 | 3 | -33 | -48 | 2 |
%F A338124 +===========================================+ (End)
%e A338124 For n=1 there are three rays that do not intersect, so a(1)=3.
%o A338124 (PARI)
%o A338124 a(n)=if( \
%o A338124 n%12==1||n%12==5,(6*n^2 + 21*n - 3)/4, \
%o A338124 n%12==2||n%12==10,(3*n^2 + 3*n + 12)/2, \
%o A338124 n%12==3,(6*n^2 + 9*n + 15)/4, \
%o A338124 n%12==6,(3*n^2 - 6*n + 48)/2, \
%o A338124 n%12==7,(6*n^2 + 21*n + 39)/4, \
%o A338124 n%12==9,(6*n^2 + 9*n + 33)/4, \
%o A338124 n%12==11,(6*n^2 + 21*n - 9)/4, \
%o A338124 n%24==4||n%24==20,(3*n^2 - 6*n + 48)/2, \
%o A338124 n%24==8||n%24==16,(3*n^2 - 6*n + 84)/2, \
%o A338124 n%120==0,(3*n^2 - 33*n - 12)/2, \
%o A338124 n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 33*n + 240)/2, \
%o A338124 n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 33*n + 276)/2, \
%o A338124 n%120==60,(3*n^2 - 33*n - 48)/2, \
%o A338124 -1);
%o A338124 vector(798, n, a(n+2))
%Y A338124 Cf. A338043 (two start points), A338122 (regions), A338123 (vertices).
%K A338124 nonn
%O A338124 1,1
%A A338124 _Lars Blomberg_, Oct 11 2020