A338122 -id:A338122 - cp's OEIS frontend

A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.

3, 4, 15, 19, 33, 31, 63, 55, 78, 82, 120, 67, 162, 154, 189, 175, 261, 217, 327, 259, 360, 370, 456, 283, 534, 514, 579, 523, 705, 619, 807, 703, 858, 874, 1008, 691, 1122, 1090, 1185, 1111, 1365, 1237, 1503, 1339, 1572, 1594, 1776, 1339, 1926, 1882, 2007, 1891

Offset: 1

Lars Blomberg, Oct 11 2020

nonn

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.

See A338122 for illustrations.

			For n=1 there are three rays that do not intersect, so a(1)=3.

Lars Blomberg, Table of n, a(n) for n = 1..800

Cf. A338042 (two start points), A338122 (regions), A338124 (edges).

a(n)=if( \
n%6==5,(3*n^2 + 10*n + 7)/4, \
n%12==1,(3*n^2 + 10*n + 11)/4, \
n%12==2||n%12==10,(3*n^2 + 28)/4, \
n%12==3,(3*n^2 + 4*n + 21)/4, \
n%12==6,(3*n^2 - 10*n + 76)/4, \
n%12==7,(3*n^2 + 10*n + 35)/4, \
n%12==9,(3*n^2 + 4*n + 33)/4, \
n%24==4||n%24==20,(3*n^2 - 12*n + 76)/4, \
n%24==8||n%24==16,(3*n^2 - 12*n + 124)/4, \
n%120==0,(3*n^2 - 40*n - 20)/4, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 40*n + 316)/4, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 40*n + 364)/4, \
n%120==60,(3*n^2 - 40*n - 68)/4, \
-1);
vector(798, n, a(n+2))

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 3 <= 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 |
+-------------------------------------------+
|               5 |   6 | 3 |  10 |   7 | 4 |
|               1 |  12 | 3 |  10 |  11 | 4 |
|           2, 10 |  12 | 3 |     |  28 | 4 |
|               3 |  12 | 3 |   4 |  21 | 4 |
|               6 |  12 | 3 | -10 |  76 | 4 |
|               7 |  12 | 3 |  10 |  35 | 4 |
|               9 |  12 | 3 |   4 |  33 | 4 |
|           4, 20 |  24 | 3 | -12 |  76 | 4 |
|           8, 16 |  24 | 3 | -12 | 124 | 4 |
|               0 | 120 | 3 | -40 | -20 | 4 |
| 12, 36, 84, 108 | 120 | 3 | -40 | 316 | 4 |
|  24, 48, 72, 96 | 120 | 3 | -40 | 364 | 4 |
|              60 | 120 | 3 | -40 | -68 | 4 |
+===========================================+ (End)

A338124 Place three points evenly spaced around 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.

Original entry on oeis.org

3, 9, 24, 36, 63, 60, 120, 114, 150, 171, 237, 138, 321, 321, 375, 378, 522, 456, 651, 564, 717, 765, 912, 606, 1068, 1059, 1158, 1116, 1413, 1284, 1614, 1482, 1716, 1791, 2019, 1470, 2247, 2229, 2373, 2322, 2736, 2544, 3009, 2796, 3147, 3249, 3558, 2802, 3858

Offset: 1

Lars Blomberg, Oct 11 2020

nonn

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.

See A338122 for illustrations.

			For n=1 there are three rays that do not intersect, so a(1)=3.

Lars Blomberg, Table of n, a(n) for n = 1..800

Cf. A338043 (two start points), A338122 (regions), A338123 (vertices).

a(n)=if( \
n%12==1||n%12==5,(6*n^2 + 21*n - 3)/4, \
n%12==2||n%12==10,(3*n^2 + 3*n + 12)/2, \
n%12==3,(6*n^2 + 9*n + 15)/4, \
n%12==6,(3*n^2 - 6*n + 48)/2, \
n%12==7,(6*n^2 + 21*n + 39)/4, \
n%12==9,(6*n^2 + 9*n + 33)/4, \
n%12==11,(6*n^2 + 21*n - 9)/4, \
n%24==4||n%24==20,(3*n^2 - 6*n + 48)/2, \
n%24==8||n%24==16,(3*n^2 - 6*n + 84)/2, \
n%120==0,(3*n^2 - 33*n - 12)/2, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 33*n + 240)/2, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 33*n + 276)/2, \
n%120==60,(3*n^2 - 33*n - 48)/2, \
-1);
vector(798, n, a(n+2))

a(n) = 4320-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 3 <= 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, 5 |  12 | 6 |  21 |  -3 | 4 |
|           2, 10 |  12 | 3 |   3 |  12 | 2 |
|               3 |  12 | 6 |   9 |  15 | 4 |
|               6 |  12 | 3 |  -6 |  48 | 2 |
|               7 |  12 | 6 |  21 |  39 | 4 |
|               9 |  12 | 6 |   9 |  33 | 4 |
|              11 |  12 | 6 |  21 |  -9 | 4 |
|           4, 20 |  24 | 3 |  -6 |  48 | 2 |
|           8, 16 |  24 | 3 |  -6 |  84 | 2 |
|               0 | 120 | 3 | -33 | -12 | 2 |
| 12, 36, 84, 108 | 120 | 3 | -33 | 240 | 2 |
|  24, 48, 72, 96 | 120 | 3 | -33 | 276 | 2 |
|              60 | 120 | 3 | -33 | -48 | 2 |
+===========================================+ (End)

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.

Original entry on oeis.org

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

Offset: 1

Lars Blomberg, Oct 26 2020

nonn

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.

			For n=1 there are four rays that do not intersect, so a(1)=1.

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

Cf. A338122, A338422 (vertices), A338423 (edges).

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))

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 |
+=================================+

cp's OEIS Frontend

A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.

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A338124 Place three points evenly spaced around 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.

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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.

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Examples

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