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A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.

%I A338123 #12 Oct 25 2020 14:17:03
%S A338123 3,4,15,19,33,31,63,55,78,82,120,67,162,154,189,175,261,217,327,259,
%T A338123 360,370,456,283,534,514,579,523,705,619,807,703,858,874,1008,691,
%U A338123 1122,1090,1185,1111,1365,1237,1503,1339,1572,1594,1776,1339,1926,1882,2007,1891
%N A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.
%C A338123 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 A338123 See A338122 for illustrations.
%H A338123 Lars Blomberg, <a href="/A338123/b338123.txt">Table of n, a(n) for n = 1..800</a>
%F A338123 a(n) = 2160-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 A338123 From _Lars Blomberg_, Oct 25 2020: (Start)
%F A338123 Conjectured for 3 <= n <= 800.
%F A338123 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F A338123 +===========================================+
%F A338123 |               r |   m | a |   b |   c | d |
%F A338123 +-------------------------------------------+
%F A338123 |               5 |   6 | 3 |  10 |   7 | 4 |
%F A338123 |               1 |  12 | 3 |  10 |  11 | 4 |
%F A338123 |           2, 10 |  12 | 3 |     |  28 | 4 |
%F A338123 |               3 |  12 | 3 |   4 |  21 | 4 |
%F A338123 |               6 |  12 | 3 | -10 |  76 | 4 |
%F A338123 |               7 |  12 | 3 |  10 |  35 | 4 |
%F A338123 |               9 |  12 | 3 |   4 |  33 | 4 |
%F A338123 |           4, 20 |  24 | 3 | -12 |  76 | 4 |
%F A338123 |           8, 16 |  24 | 3 | -12 | 124 | 4 |
%F A338123 |               0 | 120 | 3 | -40 | -20 | 4 |
%F A338123 | 12, 36, 84, 108 | 120 | 3 | -40 | 316 | 4 |
%F A338123 |  24, 48, 72, 96 | 120 | 3 | -40 | 364 | 4 |
%F A338123 |              60 | 120 | 3 | -40 | -68 | 4 |
%F A338123 +===========================================+ (End)
%e A338123 For n=1 there are three rays that do not intersect, so a(1)=3.
%o A338123 (PARI)
%o A338123 a(n)=if( \
%o A338123 n%6==5,(3*n^2 + 10*n + 7)/4, \
%o A338123 n%12==1,(3*n^2 + 10*n + 11)/4, \
%o A338123 n%12==2||n%12==10,(3*n^2 + 28)/4, \
%o A338123 n%12==3,(3*n^2 + 4*n + 21)/4, \
%o A338123 n%12==6,(3*n^2 - 10*n + 76)/4, \
%o A338123 n%12==7,(3*n^2 + 10*n + 35)/4, \
%o A338123 n%12==9,(3*n^2 + 4*n + 33)/4, \
%o A338123 n%24==4||n%24==20,(3*n^2 - 12*n + 76)/4, \
%o A338123 n%24==8||n%24==16,(3*n^2 - 12*n + 124)/4, \
%o A338123 n%120==0,(3*n^2 - 40*n - 20)/4, \
%o A338123 n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 40*n + 316)/4, \
%o A338123 n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 40*n + 364)/4, \
%o A338123 n%120==60,(3*n^2 - 40*n - 68)/4, \
%o A338123 -1);
%o A338123 vector(798, n, a(n+2))
%Y A338123 Cf. A338042 (two start points), A338122 (regions), A338124 (edges).
%K A338123 nonn
%O A338123 1,1
%A A338123 _Lars Blomberg_, Oct 11 2020