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A338422 Place four points evenly spaced on 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 A338422 #15 Nov 01 2020 08:19:31
%S A338422 4,5,24,21,64,45,96,37,152,129,216,173,304,261,384,185,488,441,600,
%T A338422 517,736,669,864,453,1016,945,1176,1053,1360,1269,1536,1025,1736,1641,
%U A338422 1944,1781,2176,2061,2400,1717,2648,2529,2904,2701,3184,3045,3456,2465,3752
%N A338422 Place four points evenly spaced on 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 A338422 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 A338422 See A338421 for illustrations.
%H A338422 Lars Blomberg, <a href="/A338422/b338422.txt">Table of n, a(n) for n = 1..642</a>
%F A338422 Conjectured for 3 <= n <= 642.
%F A338422 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F A338422 +==================================+
%F A338422 |      r |  m | a |   b |    c | d |
%F A338422 +----------------------------------+
%F A338422 |      2 |  4 | 3 |  -6 |   18 | 2 |
%F A338422 |      3 |  4 | 3 |   6 |    3 | 2 |
%F A338422 |      1 |  8 | 3 |   6 |    7 | 2 |
%F A338422 |      4 |  8 | 3 | -10 |   34 | 2 |
%F A338422 |      5 |  8 | 3 |   6 |   23 | 2 |
%F A338422 |      0 | 48 | 3 | -39 | -110 | 2 |
%F A338422 |  8, 40 | 48 | 3 | -39 |  194 | 2 |
%F A338422 | 16, 32 | 48 | 3 | -39 |  226 | 2 |
%F A338422 |     24 | 48 | 3 | -39 |  114 | 2 |
%F A338422 +==================================+
%e A338422 For n=1 there are four rays that do not intersect, so a(1)=4.
%o A338422 (PARI)
%o A338422 a(n)={ if(
%o A338422 n==1, 4,
%o A338422 n==2, 5,
%o A338422 n%4==2, (3*n^2 - 6*n + 18)/2,
%o A338422 n%4==3, (3*n^2 + 6*n + 3)/2,
%o A338422 n%8==1, (3*n^2 + 6*n + 7)/2,
%o A338422 n%8==4, (3*n^2 - 10*n + 34)/2,
%o A338422 n%8==5, (3*n^2 + 6*n + 23)/2,
%o A338422 n%48==0, (3*n^2 - 39*n - 110)/2,
%o A338422 n%48==8||n%48==40, (3*n^2 - 39*n + 194)/2,
%o A338422 n%48==16||n%48==32, (3*n^2 - 39*n + 226)/2,
%o A338422 n%48==24, (3*n^2 - 39*n + 114)/2,
%o A338422 -1); }
%o A338422 vector(642, n, a(n))
%Y A338422 Cf. A338123, A338421 (regions), A338423 (edges).
%K A338422 nonn
%O A338422 1,1
%A A338422 _Lars Blomberg_, Oct 26 2020