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Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.
4

%I #18 Dec 10 2022 10:46:31

%S 1,4,10,18,28,40,54,70,88,108,130,154,180,208,238,270,304,340,378,418,

%T 460,504,550,598,648,700,754,810,868,928,990,1054,1120,1188,1258,1330,

%U 1404,1480,1558,1638,1720,1804,1890,1978,2068,2160,2254,2350,2448,2548

%N Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

%C For n > 3 same as A028552(n-3).

%F a(n) = n*(n-3) for n > 3.

%F a(n) = A217745(n) - A217746(n).

%F From _Amiram Eldar_, Dec 10 2022: (Start)

%F Sum_{n>=3} 1/a(n) = 29/18.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 23/18 - 2*log(2)/3. (End)

%e a(3) = 1 since the equilateral triangle has no diagonals and therefore one exterior region with infinite area.

%e a(4) = 4 since the two diagonals of the square divide the exterior in four regions with infinite area.

%e a(5) = 10 since the ten diagonals of the regular pentagon divide the exterior in ten regions with infinite area of two different shapes.

%t a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* _Amiram Eldar_, Dec 10 2022 *)

%o (PARI) a(n) = if(n == 3, 1, n*(n-3)); \\ _Amiram Eldar_, Dec 10 2022

%Y Cf. A004526, A007678, A028552, A164004, A217745, A217746.

%K nonn

%O 3,2

%A _Martin Renner_, Mar 23 2013