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 A338421 Place four points evenly spaced on 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. 3
 1, 4, 21, 16, 57, 44, 93, 36, 149, 132, 217, 176, 301, 268, 385, 208, 489, 452, 605, 528, 737, 684, 869, 532, 1021, 964, 1185, 1072, 1365, 1292, 1545, 1112, 1745, 1668, 1957, 1808, 2185, 2092, 2413, 1844, 2661, 2564, 2921, 2736, 3197, 3084, 3473, 2696, 3769 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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..642 Lars Blomberg, Illustration for n=3 Lars Blomberg, Illustration for n=7 Lars Blomberg, Illustration for n=8 Lars Blomberg, Illustration for n=16 Lars Blomberg, Illustration for n=22 Lars Blomberg, Illustration for n=26 Lars Blomberg, Illustration for n=27 Lars Blomberg, Illustration for n=38 FORMULA Conjectured for 3 <= n <= 642. 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 | +---------------------------------+ |      2 |  4 | 3 |  -4 |   4 | 2 | |      1 |  8 | 3 |   7 |  -8 | 2 | |      3 |  8 | 3 |   7 |  -6 | 2 | |      4 |  8 | 3 |  -8 |  16 | 2 | |      5 |  8 | 3 |   7 |   4 | 2 | |      7 |  8 | 3 |   7 | -10 | 2 | |      0 | 48 | 3 | -31 | -32 | 2 | |  8, 40 | 48 | 3 | -31 | 128 | 2 | | 16, 32 | 48 | 3 | -31 | 144 | 2 | |     24 | 48 | 3 | -31 |  80 | 2 | +=================================+ EXAMPLE For n=1 there are four rays that do not intersect, so a(1)=1. PROG (PARI) a(n)={if( n==1, 1, n==2, 4, n%4==2, (3*n^2 - 4*n + 4)/2, n%8==1, (3*n^2 + 7*n - 8)/2, n%8==3, (3*n^2 + 7*n - 6)/2, n%8==4, (3*n^2 - 8*n + 16)/2, n%8==5, (3*n^2 + 7*n + 4)/2, n%8==7, (3*n^2 + 7*n - 10)/2, n%48==0, (3*n^2 - 31*n - 32)/2, n%48==8||n%48==40, (3*n^2 - 31*n + 128)/2, n%48==16||n%48==32, (3*n^2 - 31*n + 144)/2, n%48==24, (3*n^2 - 31*n + 80)/2, -1); } vector(642, n, a(n)) CROSSREFS Cf. A338122, A338422 (vertices), A338423 (edges). Sequence in context: A333433 A202450 A144292 * A329404 A076943 A138228 Adjacent sequences:  A338418 A338419 A338420 * A338422 A338423 A338424 KEYWORD nonn AUTHOR Lars Blomberg, Oct 26 2020 STATUS approved

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Last modified April 11 14:58 EDT 2021. Contains 342886 sequences. (Running on oeis4.)