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 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. 4
 1, 6, 10, 18, 31, 30, 58, 60, 73, 90, 118, 72, 160, 168, 187, 204, 262, 240, 325, 306, 358, 396, 457, 324, 535, 546, 580, 594, 709, 666, 808, 780, 859, 918, 1012, 780, 1126, 1140, 1189, 1212, 1372, 1308, 1507, 1458, 1576, 1656, 1783, 1464, 1933, 1950, 2014, 2034 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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. LINKS Lars Blomberg, Table of n, a(n) for n = 1..800 Lars Blomberg, Illustration for n=3 Lars Blomberg, Illustration for n=9 Lars Blomberg, Illustration for n=15 Lars Blomberg, Illustration for n=24 Lars Blomberg, Illustration for n=27 Lars Blomberg, Illustration for n=30 Lars Blomberg, Illustration for n=45 FORMULA 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) From Lars Blomberg, Oct 25 2020: (Start) Conjectured for 1 <= n <= 800. Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d. +===========================================+ |               r |   m | a |   b |   c | d | +-------------------------------------------+ |               1 |  12 | 3 |  11 | -10 | 4 | |           2, 10 |  12 | 3 |   6 |     | 4 | |               3 |  12 | 3 |   5 |  -2 | 4 | |               5 |  12 | 3 |  11 |  -6 | 4 | |               6 |  12 | 3 |  -2 |  24 | 4 | |               7 |  12 | 3 |  11 |   8 | 4 | |               9 |  12 | 3 |   5 |   4 | 4 | |              11 |  12 | 3 |  11 | -12 | 4 | |           4, 20 |  24 | 3 |     |  24 | 4 | |           8, 16 |  24 | 3 |     |  48 | 4 | |               0 | 120 | 3 | -26 |     | 4 | | 12, 36, 84, 108 | 120 | 3 | -26 | 168 | 4 | |  24, 48, 72, 96 | 120 | 3 | -26 | 192 | 4 | |              60 | 120 | 3 | -26 | -24 | 4 | +===========================================+ (End) EXAMPLE For n=1 there are three rays that do not intersect, so a(1)=1. PROG (PARI) a(n)=if( \ n%12==1, (3*n^2 + 11*n - 10)/4, \ n%12==2||n%12==10, (3*n^2 + 6*n)/4, \ n%12==3, (3*n^2 + 5*n - 2)/4, \ n%12==5, (3*n^2 + 11*n - 6)/4, \ n%12==6, (3*n^2 - 2*n + 24)/4, \ n%12==7, (3*n^2 + 11*n + 8)/4, \ n%12==9, (3*n^2 + 5*n + 4)/4, \ n%12==11, (3*n^2 + 11*n - 12)/4, \ n%24==4||n%24==20, (3*n^2 + 24)/4, \ n%24==8||n%24==16, (3*n^2 + 48)/4, \ n%120==0, (3*n^2 - 26*n)/4, \ n%120==12||n%120==36||n%120==84||n%120==108, (3*n^2 - 26*n + 168)/4, \ n%120==24||n%120==48||n%120==72||n%120==96, (3*n^2 - 26*n + 192)/4, \ n%120==60, (3*n^2 - 26*n - 24)/4, \ -1); vector(800, n, a(n)) CROSSREFS Cf. A338041 (two start points), A338123 (vertices), A338124 (edges). Sequence in context: A331861 A032641 A293555 * A169873 A079471 A134351 Adjacent sequences:  A338119 A338120 A338121 * A338123 A338124 A338125 KEYWORD nonn AUTHOR Lars Blomberg, Oct 11 2020 STATUS approved

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Last modified June 12 09:24 EDT 2021. Contains 344946 sequences. (Running on oeis4.)