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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.

%I A338041 #21 Oct 19 2020 16:41:57
%S A338041 1,2,7,6,15,12,25,20,37,30,51,42,67,56,85,72,105,90,127,110,151,132,
%T A338041 177,156,205,182,235,210,267,240,301,272,337,306,375,342,415,380,457,
%U A338041 420,501,462,547,506,595,552,645,600,697,650,751,702,807,756,865,812,925
%N 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.
%C A338041 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.
%C A338041 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.
%H A338041 Lars Blomberg, <a href="/A338041/a338041.png">Illustration for n = 3</a>
%H A338041 Lars Blomberg, <a href="/A338041/a338041_1.png">Illustration for n = 6</a>
%H A338041 Lars Blomberg, <a href="/A338041/a338041_2.png">Illustration for n = 7</a>
%H A338041 Lars Blomberg, <a href="/A338041/a338041_3.png">Illustration for n = 18</a>
%H A338041 Lars Blomberg, <a href="/A338041/a338041_4.png">Illustration for n = 19</a>
%H A338041 Lars Blomberg, <a href="/A338041/a338041_5.png">Illustration for n = 33</a>
%F A338041 a(n) = (n^2 + 8*n - 5)/4, n odd; (n^2 + 2*n)/4, n even (conjectured).
%F A338041 Conjectured by _Stefano Spezia_, Oct 08 2020 after _Lars Blomberg_: (Start)
%F A338041 G.f.: x*(1 + x + 3*x^2 - 3*x^3)/((1 - x)^3*(1 + x)^2).
%F A338041 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
%e A338041 For n=1:    <-----x     x----->   so a(1)=1.
%e A338041 For n=2:    <-----x<--->x----->   so a(2)=2.
%o A338041 (PARI) a(n)=if(n%2==1,(n^2 + 8*n - 5)/4,(n^2 + 2*n)/4);
%o A338041 vector(200, n, a(n))
%Y A338041 Cf. A338042 (vertices), A338043 (edges).
%K A338041 nonn
%O A338041 1,2
%A A338041 _Lars Blomberg_, Oct 08 2020