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
KEYWORD
nonn
AUTHOR
Lars Blomberg, Oct 11 2020
STATUS
approved