login
A338422
Place four points evenly spaced on 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, 5, 24, 21, 64, 45, 96, 37, 152, 129, 216, 173, 304, 261, 384, 185, 488, 441, 600, 517, 736, 669, 864, 453, 1016, 945, 1176, 1053, 1360, 1269, 1536, 1025, 1736, 1641, 1944, 1781, 2176, 2061, 2400, 1717, 2648, 2529, 2904, 2701, 3184, 3045, 3456, 2465, 3752
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 | -6 | 18 | 2 |
| 3 | 4 | 3 | 6 | 3 | 2 |
| 1 | 8 | 3 | 6 | 7 | 2 |
| 4 | 8 | 3 | -10 | 34 | 2 |
| 5 | 8 | 3 | 6 | 23 | 2 |
| 0 | 48 | 3 | -39 | -110 | 2 |
| 8, 40 | 48 | 3 | -39 | 194 | 2 |
| 16, 32 | 48 | 3 | -39 | 226 | 2 |
| 24 | 48 | 3 | -39 | 114 | 2 |
+==================================+
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, 5,
n%4==2, (3*n^2 - 6*n + 18)/2,
n%4==3, (3*n^2 + 6*n + 3)/2,
n%8==1, (3*n^2 + 6*n + 7)/2,
n%8==4, (3*n^2 - 10*n + 34)/2,
n%8==5, (3*n^2 + 6*n + 23)/2,
n%48==0, (3*n^2 - 39*n - 110)/2,
n%48==8||n%48==40, (3*n^2 - 39*n + 194)/2,
n%48==16||n%48==32, (3*n^2 - 39*n + 226)/2,
n%48==24, (3*n^2 - 39*n + 114)/2,
-1); }
vector(642, n, a(n))
CROSSREFS
Cf. A338123, A338421 (regions), A338423 (edges).
Sequence in context: A341586 A010302 A361981 * A171885 A331261 A063986
KEYWORD
nonn
AUTHOR
Lars Blomberg, Oct 26 2020
STATUS
approved