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%I #12 Nov 01 2020 08:19:21
%S 4,8,44,36,120,88,188,72,300,260,432,348,604,528,768,392,976,892,1204,
%T 1044,1472,1352,1732,984,2036,1908,2360,2124,2724,2560,3080,2136,3480,
%U 3308,3900,3588,4360,4152,4812,3560,5308,5092,5824,5436,6380,6128,6928,5160
%N 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 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 See A338421 for illustrations.
%H Lars Blomberg, <a href="/A338423/b338423.txt">Table of n, a(n) for n = 1..642</a>
%F Conjectured for 3 <= n <= 642.
%F Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
%F +=================================+
%F | r | m | a | b | c | d |
%F +---------------------------------+
%F | 2 | 4 | 3 | -5 | 10 | 1 |
%F | 1 | 8 | 6 | 13 | -3 | 2 |
%F | 3 | 8 | 6 | 13 | -5 | 2 |
%F | 4 | 8 | 3 | -9 | 24 | 1 |
%F | 5 | 8 | 6 | 13 | 25 | 2 |
%F | 7 | 8 | 6 | 13 | -9 | 2 |
%F | 0 | 48 | 3 | -35 | -72 | 1 |
%F | 8, 40 | 48 | 3 | -35 | 160 | 1 |
%F | 16, 32 | 48 | 3 | -35 | 184 | 1 |
%F | 24 | 48 | 3 | -35 | 96 | 1 |
%F +=================================+ (End)
%e For n=1 there are four rays that do not intersect, so a(1)=4.
%o (PARI)
%o a(n)={if(
%o n==1,4,
%o n==2,8,
%o n%4==2,3*n^2 - 5*n + 10,
%o n%8==1,(6*n^2 + 13*n - 3)/2,
%o n%8==3,(6*n^2 + 13*n - 5)/2,
%o n%8==4,3*n^2 - 9*n + 24,
%o n%8==5,(6*n^2 + 13*n + 25)/2,
%o n%8==7,(6*n^2 + 13*n - 9)/2,
%o n%48==0,3*n^2 - 35*n - 72,
%o n%48==8||n%48==40,3*n^2 - 35*n + 160,
%o n%48==16||n%48==32,3*n^2 - 35*n + 184,
%o n%48==24,3*n^2 - 35*n + 96,
%o -1);}
%o vector(642, n, a(n))
%Y Cf. A338124, A338421 (regions), A338422 (vertices).
%K nonn
%O 1,1
%A _Lars Blomberg_, Oct 26 2020