This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A338122 #14 Oct 25 2020 14:16:48 %S A338122 1,6,10,18,31,30,58,60,73,90,118,72,160,168,187,204,262,240,325,306, %T A338122 358,396,457,324,535,546,580,594,709,666,808,780,859,918,1012,780, %U A338122 1126,1140,1189,1212,1372,1308,1507,1458,1576,1656,1783,1464,1933,1950,2014,2034 %N A338122 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details. %C A338122 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 A338122 To produce the illustrations below, all pairwise intersections between the rays are calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit. %H A338122 Lars Blomberg, <a href="/A338122/b338122.txt">Table of n, a(n) for n = 1..800</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122.png">Illustration for n=3</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_1.png">Illustration for n=9</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_2.png">Illustration for n=15</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_3.png">Illustration for n=24</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_4.png">Illustration for n=27</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_5.png">Illustration for n=30</a> %H A338122 Lars Blomberg, <a href="/A338122/a338122_6.png">Illustration for n=45</a> %F A338122 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 A338122 From _Lars Blomberg_, Oct 25 2020: (Start) %F A338122 Conjectured for 1 <= n <= 800. %F A338122 Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d. %F A338122 +===========================================+ %F A338122 | r | m | a | b | c | d | %F A338122 +-------------------------------------------+ %F A338122 | 1 | 12 | 3 | 11 | -10 | 4 | %F A338122 | 2, 10 | 12 | 3 | 6 | | 4 | %F A338122 | 3 | 12 | 3 | 5 | -2 | 4 | %F A338122 | 5 | 12 | 3 | 11 | -6 | 4 | %F A338122 | 6 | 12 | 3 | -2 | 24 | 4 | %F A338122 | 7 | 12 | 3 | 11 | 8 | 4 | %F A338122 | 9 | 12 | 3 | 5 | 4 | 4 | %F A338122 | 11 | 12 | 3 | 11 | -12 | 4 | %F A338122 | 4, 20 | 24 | 3 | | 24 | 4 | %F A338122 | 8, 16 | 24 | 3 | | 48 | 4 | %F A338122 | 0 | 120 | 3 | -26 | | 4 | %F A338122 | 12, 36, 84, 108 | 120 | 3 | -26 | 168 | 4 | %F A338122 | 24, 48, 72, 96 | 120 | 3 | -26 | 192 | 4 | %F A338122 | 60 | 120 | 3 | -26 | -24 | 4 | %F A338122 +===========================================+ (End) %e A338122 For n=1 there are three rays that do not intersect, so a(1)=1. %o A338122 (PARI) %o A338122 a(n)=if( \ %o A338122 n%12==1,(3*n^2 + 11*n - 10)/4, \ %o A338122 n%12==2||n%12==10,(3*n^2 + 6*n)/4, \ %o A338122 n%12==3,(3*n^2 + 5*n - 2)/4, \ %o A338122 n%12==5,(3*n^2 + 11*n - 6)/4, \ %o A338122 n%12==6,(3*n^2 - 2*n + 24)/4, \ %o A338122 n%12==7,(3*n^2 + 11*n + 8)/4, \ %o A338122 n%12==9,(3*n^2 + 5*n + 4)/4, \ %o A338122 n%12==11,(3*n^2 + 11*n - 12)/4, \ %o A338122 n%24==4||n%24==20,(3*n^2 + 24)/4, \ %o A338122 n%24==8||n%24==16,(3*n^2 + 48)/4, \ %o A338122 n%120==0,(3*n^2 - 26*n)/4, \ %o A338122 n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 26*n + 168)/4, \ %o A338122 n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 26*n + 192)/4, \ %o A338122 n%120==60,(3*n^2 - 26*n - 24)/4, \ %o A338122 -1); %o A338122 vector(800, n, a(n)) %Y A338122 Cf. A338041 (two start points), A338123 (vertices), A338124 (edges). %K A338122 nonn %O A338122 1,2 %A A338122 _Lars Blomberg_, Oct 11 2020