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

3,4

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

Links

Table of n, a(n) for n=3..62.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A062299, A062300.

Sequence in context: A375581 A320848 A228234 * A060526 A301599 A036408

Adjacent sequences: A360789 A360790 A360791 * A360793 A360794 A360795

Keyword

nonn

Author

A. Timothy Royappa, Feb 20 2023

Extensions

More terms from Andrew Howroyd, Feb 20 2023

Status

approved