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A217748
Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.
4
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
3,2
COMMENTS
For n > 3 same as A028552(n-3).
FORMULA
a(n) = n*(n-3) for n > 3.
a(n) = A217745(n) - A217746(n).
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)
EXAMPLE
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.
MATHEMATICA
a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* Amiram Eldar, Dec 10 2022 *)
PROG
(PARI) a(n) = if(n == 3, 1, n*(n-3)); \\ Amiram Eldar, Dec 10 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Martin Renner, Mar 23 2013
STATUS
approved