login
A338423
Place four points evenly spaced on 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.
3
4, 8, 44, 36, 120, 88, 188, 72, 300, 260, 432, 348, 604, 528, 768, 392, 976, 892, 1204, 1044, 1472, 1352, 1732, 984, 2036, 1908, 2360, 2124, 2724, 2560, 3080, 2136, 3480, 3308, 3900, 3588, 4360, 4152, 4812, 3560, 5308, 5092, 5824, 5436, 6380, 6128, 6928, 5160
OFFSET
1,1
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.
See A338421 for illustrations.
LINKS
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 | -5 | 10 | 1 |
| 1 | 8 | 6 | 13 | -3 | 2 |
| 3 | 8 | 6 | 13 | -5 | 2 |
| 4 | 8 | 3 | -9 | 24 | 1 |
| 5 | 8 | 6 | 13 | 25 | 2 |
| 7 | 8 | 6 | 13 | -9 | 2 |
| 0 | 48 | 3 | -35 | -72 | 1 |
| 8, 40 | 48 | 3 | -35 | 160 | 1 |
| 16, 32 | 48 | 3 | -35 | 184 | 1 |
| 24 | 48 | 3 | -35 | 96 | 1 |
+=================================+ (End)
EXAMPLE
For n=1 there are four rays that do not intersect, so a(1)=4.
PROG
(PARI)
a(n)={if(
n==1, 4,
n==2, 8,
n%4==2, 3*n^2 - 5*n + 10,
n%8==1, (6*n^2 + 13*n - 3)/2,
n%8==3, (6*n^2 + 13*n - 5)/2,
n%8==4, 3*n^2 - 9*n + 24,
n%8==5, (6*n^2 + 13*n + 25)/2,
n%8==7, (6*n^2 + 13*n - 9)/2,
n%48==0, 3*n^2 - 35*n - 72,
n%48==8||n%48==40, 3*n^2 - 35*n + 160,
n%48==16||n%48==32, 3*n^2 - 35*n + 184,
n%48==24, 3*n^2 - 35*n + 96,
-1); }
vector(642, n, a(n))
CROSSREFS
Cf. A338124, A338421 (regions), A338422 (vertices).
Sequence in context: A137052 A111411 A137054 * A163343 A284972 A045639
KEYWORD
nonn
AUTHOR
Lars Blomberg, Oct 26 2020
STATUS
approved