A360792 Integer portion of area of inscribed circle in a regular polygon having n sides of unit length.

0, 0, 1, 2, 3, 4, 5, 7, 9, 10, 12, 15, 17, 19, 22, 25, 28, 31, 34, 37, 41, 45, 49, 53, 57, 61, 66, 71, 75, 80, 86, 91, 96, 102, 108, 114, 120, 126, 133, 139, 146, 153, 160, 167, 175, 182, 190, 198, 206, 214, 223, 231, 240, 249, 258, 267, 276, 285, 295, 305

Offset: 3

Author

A. Timothy Royappa, Feb 20 2023

Keywords

• nonn

Comments

Asymptotically equivalent to the area of a circle with circumference n, which is Pi*r^2 with r = n/(2*Pi), thus a(n) ~ n^2 / (4*Pi). (The same follows from taking tan(x) = x in the FORMULA.) Indeed, the integer part of this expression is equal to a(n) or a(n)+1, for all n. - M. F. Hasler, Apr 03 2025

Examples

			For n = 5, the circle inscribed in a regular pentagon with sides of unit length has area (Pi/4)*cot(Pi/5)^2 = 1.4878796365..., so a(5) = floor(1.4878796365...) = 1.
		

Crossrefs

Cf. A062299, A062300.

Programs

Maple

a:= n-> floor(Pi/(2*tan(Pi/n))^2):
seq(a(n), n=3..65);  # Alois P. Heinz, Feb 20 2023

Mathematica

a[n_] := Floor[(Pi/4)*Cot[Pi/n]^2]; Array[a, 60, 3] (* Amiram Eldar, Feb 24 2023 *)

PARI

a(n) = floor((Pi/4)/tan(Pi/n)^2) \\ Andrew Howroyd, Feb 20 2023

PARI

apply( {A360792(n)=Pi/4\tan(Pi/n)^2}, [3..62]) \\ M. F. Hasler, Apr 03 2025

Formula

a(n) = floor((Pi/4)*(cot(Pi/n)^2)).

Extensions

More terms from Andrew Howroyd, Feb 20 2023