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

3,2

Comments

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

Links

Table of n, a(n) for n=3..52.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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)

From Stefano Spezia, Jul 06 2026: (Start)

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

E.g.f.: x*(12 + 6*x + x^2 + 6*exp(x)*(x - 2))/6. (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

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

Sequence in context: A009876 A161958 A013921 * A028552 A009877 A009880

Adjacent sequences: A217745 A217746 A217747 * A217749 A217750 A217751

Keyword

nonn,easy

Author

Martin Renner, Mar 23 2013

Status

approved