A374296 a(n) is the integer part of the area of a regular n-gon whose side lengths are n.

3, 16, 43, 93, 178, 309, 500, 769, 1133, 1612, 2228, 3005, 3969, 5147, 6570, 8268, 10275, 12627, 15360, 18514, 22130, 26250, 30921, 36187, 42099, 48707, 56063, 64221, 73239, 83174, 94087, 106039, 119095, 133320, 148782, 165551, 183699, 203299

Offset

3,1

Links

Paolo Xausa, Table of n, a(n) for n = 3..10000

Nicolay Avilov, Illustration of a(3)-a(7)

Formula

a(n) = floor(n^3/(4*tan(Pi/n))).

a(n) = n^4/(4*Pi) - (Pi/12)*n^2 + O(1). - Charles R Greathouse IV, Jul 03 2024

Example

Areas of polygons (starting from n=3):
  3.897... (equilateral triangle), so a(3) = 3,
 16.000... (square), so a(4) = 16,
 43.011... (pentagon), so a(5) = 43,
 93.530... (hexagon), so a(6) = 93,
178.061... (heptagon), so a(7) = 178.

Mathematica

A374296[n_] := Floor[n^3*Cot[Pi/n]/4]; Array[A374296, 50, 3] (* Paolo Xausa, Sep 29 2025 *)

Prog

(PARI) a(n)=n^3\(4*tan(Pi/n)) \\ Charles R Greathouse IV, Sep 29 2025

Crossrefs

Cf. A064313.

Sequence in context: A271374 A147874 A092466 * A152618 A296947 A255211

Adjacent sequences: A374293 A374294 A374295 * A374297 A374298 A374299

Keyword

nonn

Author

Nicolay Avilov, Jul 03 2024

Status

approved