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
KEYWORD
nonn
AUTHOR
Lars Blomberg, Oct 26 2020
STATUS
approved