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A338041
Draw n rays from each of two distinct points in the plane; a(n) is the number of regions thus created. See Comments for details.
5
1, 2, 7, 6, 15, 12, 25, 20, 37, 30, 51, 42, 67, 56, 85, 72, 105, 90, 127, 110, 151, 132, 177, 156, 205, 182, 235, 210, 267, 240, 301, 272, 337, 306, 375, 342, 415, 380, 457, 420, 501, 462, 547, 506, 595, 552, 645, 600, 697, 650, 751, 702, 807, 756, 865, 812, 925
OFFSET
1,2
COMMENTS
The rays are evenly spaced around each point. The first ray of one point goes opposite to the direction to the other point. Should a ray hit the other point it terminates there, that is, it is converted to a line segment.
To produce the illustrations below, all pairwise intersections between the rays is 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.
FORMULA
a(n) = (n^2 + 8*n - 5)/4, n odd; (n^2 + 2*n)/4, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: x*(1 + x + 3*x^2 - 3*x^3)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
EXAMPLE
For n=1: <-----x x-----> so a(1)=1.
For n=2: <-----x<--->x-----> so a(2)=2.
PROG
(PARI) a(n)=if(n%2==1, (n^2 + 8*n - 5)/4, (n^2 + 2*n)/4);
vector(200, n, a(n))
CROSSREFS
Cf. A338042 (vertices), A338043 (edges).
Sequence in context: A089417 A168205 A265495 * A365007 A082017 A072985
KEYWORD
nonn
AUTHOR
Lars Blomberg, Oct 08 2020
STATUS
approved