A338042 - OEIS

%I #23 Oct 19 2020 16:41:48

%S 2,2,4,2,8,4,14,8,22,14,32,22,44,32,58,44,74,58,92,74,112,92,134,112,

%T 158,134,184,158,212,184,242,212,274,242,308,274,344,308,382,344,422,

%U 382,464,422,508,464,554,508,602,554,652,602,704,652,758,704,814,758

%N Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

%C 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 See A338041 for illustrations.

%F a(n) = (n^2 + 7)/4, n odd; (n^2 - 6*n + 16)/4, n even (conjectured).

%F Conjectured by _Stefano Spezia_, Oct 08 2020 after _Lars Blomberg_: (Start)

%F G.f.: 2*x*(1 - x^2 - x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2).

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)

%F _Hugo Pfoertner_, Oct 08 2020: Apparently a(n)=2*(A008795(n-3)+1).

%e For n=1:    <-----x     x----->   so a(1)=2.

%e For n=2:    <-----x<--->x----->   so a(2)=2.

%o (PARI) a(n)=if(n%2==1,(n^2 + 7)/4,(n^2 - 6*n + 16)/4)

%o vector(200, n, a(n))

%Y Cf. A338041 (regions), A338043 (edges), A008795.

%K nonn

%O 1,1

%A _Lars Blomberg_, Oct 08 2020