A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.
1, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548
Offset: 3
Keywords
Examples
a(3) = 1 since the equilateral triangle has no diagonals and therefore one exterior region with infinite area. a(4) = 4 since the two diagonals of the square divide the exterior in four regions with infinite area. a(5) = 10 since the ten diagonals of the regular pentagon divide the exterior in ten regions with infinite area of two different shapes.
Programs
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Mathematica
a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* Amiram Eldar, Dec 10 2022 *)
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PARI
a(n) = if(n == 3, 1, n*(n-3)); \\ Amiram Eldar, Dec 10 2022
Formula
a(n) = n*(n-3) for n > 3.
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=3} 1/a(n) = 29/18.
Sum_{n>=3} (-1)^(n+1)/a(n) = 23/18 - 2*log(2)/3. (End)
Comments