A369801 Maximum number of segments between n points on a circle so that they can be colored in 2 colors so that each of them intersects (at an internal point) at most one other segment of the same color.
1, 3, 6, 10, 15, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, 64, 67, 72, 75, 80, 83, 88, 91, 96, 99, 104, 107, 112, 115, 120, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 171, 176, 179, 184, 187, 192, 195, 200, 203, 208, 211, 216, 219, 224, 227
Offset: 2
Links
- Bulgarian Winter Mathematical Competition "Atanas Radev", Problems and solutions brochure, Problem 9.4, p. 6 (in Bulgarian).
- Art of Problem Solving, High School Olympiads - Colored segments, 2024.
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Programs
-
Mathematica
Drop[CoefficientList[Series[ -x^2*(x^7-2*x^4-2*x^3-2*x^2-2*x-1)/((x+1)*(x-1)^2),{x,0,59}],x],2] (* James C. McMahon, Mar 08 2024 *) -
Python
def A369801(n): return (n-2<<2)-(n&1) if n>=7 else (1, 3, 6, 10, 15)[n-2] # Chai Wah Wu, Mar 30 2024
Formula
a(n) = n*(n-1)/2 for n<=6, a(2*k+1) = 8*k-5 if k>=3, a(2*k) = 8*k-8 if k>=4.
G.f.: -x^2*(x^7-2*x^4-2*x^3-2*x^2-2*x-1)/((x+1)*(x-1)^2).