A062300 -id:A062300 - cp's OEIS frontend

A032615 a(n) = floor(n/Pi).

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24

Offset: 0

Patrick De Geest, May 15 1998

a(n) = A062300(n-1) = floor(1/sin(Pi/n)) for almost all indices, the next exception after n = 6 being n = 80143857, cf. link to Kevin Ryde's post to the SeqFan list. - M. F. Hasler, Oct 20 2016

R. Israel, in reply to K. Ryde, Re: nearly equal floor(n/Pi) A032615 and A062300, SeqFan list, Oct 19 2016.
Index entries for sequences related to Beatty sequences

Cf. A062300. Beatty seq. of A049541.

A032615:=n->floor(n/Pi); seq(A032615(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/Pi], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)

a(n)=n\Pi \\ Charles R Greathouse IV, Dec 10 2013

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

Original entry on oeis.org

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

A. Timothy Royappa, Feb 20 2023

nonn

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

			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.

Cf. A062299, A062300.

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

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

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

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

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

More terms from Andrew Howroyd, Feb 20 2023

cp's OEIS Frontend

A032615 a(n) = floor(n/Pi).

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

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