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A338423 Place four points evenly spaced on 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 A338423 #15 Aug 17 2025 01:25:48
%S A338423 4,8,44,36,120,88,188,72,300,260,432,348,604,528,768,392,976,892,1204,
%T A338423 1044,1472,1352,1732,984,2036,1908,2360,2124,2724,2560,3080,2136,3480,
%U A338423 3308,3900,3588,4360,4152,4812,3560,5308,5092,5824,5436,6380,6128,6928,5160
%N A338423 Place four points evenly spaced on 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 A338423 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 A338423 See A338421 for illustrations.
%H A338423 Lars Blomberg, <a href="/A338423/b338423.txt">Table of n, a(n) for n = 1..642</a>
%F A338423 Conjectured for 3 <= n <= 642.
%F A338423 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F A338423 +=================================+
%F A338423 |      r |  m | a |   b |   c | d |
%F A338423 +---------------------------------+
%F A338423 |      2 |  4 | 3 |  -5 |  10 | 1 |
%F A338423 |      1 |  8 | 6 |  13 |  -3 | 2 |
%F A338423 |      3 |  8 | 6 |  13 |  -5 | 2 |
%F A338423 |      4 |  8 | 3 |  -9 |  24 | 1 |
%F A338423 |      5 |  8 | 6 |  13 |  25 | 2 |
%F A338423 |      7 |  8 | 6 |  13 |  -9 | 2 |
%F A338423 |      0 | 48 | 3 | -35 | -72 | 1 |
%F A338423 |  8, 40 | 48 | 3 | -35 | 160 | 1 |
%F A338423 | 16, 32 | 48 | 3 | -35 | 184 | 1 |
%F A338423 |     24 | 48 | 3 | -35 |  96 | 1 |
%F A338423 +=================================+
%e A338423 For n=1 there are four rays that do not intersect, so a(1)=4.
%o A338423 (PARI)
%o A338423 a(n)={if(
%o A338423 n==1,4,
%o A338423 n==2,8,
%o A338423 n%4==2,3*n^2 - 5*n + 10,
%o A338423 n%8==1,(6*n^2 + 13*n - 3)/2,
%o A338423 n%8==3,(6*n^2 + 13*n - 5)/2,
%o A338423 n%8==4,3*n^2 - 9*n + 24,
%o A338423 n%8==5,(6*n^2 + 13*n + 25)/2,
%o A338423 n%8==7,(6*n^2 + 13*n - 9)/2,
%o A338423 n%48==0,3*n^2 - 35*n - 72,
%o A338423 n%48==8||n%48==40,3*n^2 - 35*n + 160,
%o A338423 n%48==16||n%48==32,3*n^2 - 35*n + 184,
%o A338423 n%48==24,3*n^2 - 35*n + 96,
%o A338423 -1);}
%o A338423 vector(642, n, a(n))
%Y A338423 Cf. A338124, A338421 (regions), A338422 (vertices).
%K A338423 nonn
%O A338423 1,1
%A A338423 _Lars Blomberg_, Oct 26 2020