A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

2, 5, 9, 8, 0, 7, 6, 2, 1, 1, 3, 5, 3, 3, 1, 5, 9, 4, 0, 2, 9, 1, 1, 6, 9, 5, 1, 2, 2, 5, 8, 8, 0, 8, 5, 5, 0, 4, 1, 4, 2, 0, 7, 8, 8, 0, 7, 1, 5, 5, 7, 0, 9, 4, 2, 0, 8, 3, 7, 1, 0, 4, 6, 9, 1, 7, 7, 8, 9, 9, 5, 2, 5, 3, 6, 3, 2, 0, 0, 0, 5, 5, 6, 2, 1, 7, 1, 9, 2, 8, 0, 1, 3, 5, 8, 7, 2, 8, 6, 3, 5, 1, 3, 4, 3

Offset

1,1

Comments

Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.

Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - R. J. Mathar, Jul 26 2007

This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - Jean-François Alcover, Dec 19 2016

Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - Charles R Greathouse IV, May 21 2022

For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.

Kiran S. Kedlaya, A < B, (1999), Problem 6.1, p. 6.

J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.

Michael Penn, not as bad as it seems..., YouTube video, 2021.

Eric Weisstein et al., Root of Unity

Eric Weisstein's World of Mathematics, de Moivre Number

Eric Weisstein's World of Mathematics, Twenty-Vertex Entropy Constant

Wikipedia, Hexagon

Wikipedia, Regular polygon

Index entries for algebraic numbers, degree 2

Formula

Equals (3*sqrt(3))/2, that is, 2*A104954.

Equals Product_{k>=3} (((k - 1)^2*(k + 2))/((k + 1)^2*(k - 2)))^(k/2). - Antonio Graciá Llorente, Oct 13 2024

Example

2.59807621135331594029116951225880855041420788071557094208371046917789952536320...

Mathematica

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

Prog

(PARI) 3*sqrt(3)/2 \\ G. C. Greubel, Jul 03 2017

Crossrefs

Cf. A002194, A104954, A104955, A104957.

Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.

Sequence in context: A089578 A020852 A053477 * A360897 A020820 A111290

Adjacent sequences: A104953 A104954 A104955 * A104957 A104958 A104959

Keyword

nonn,cons,changed

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Status

approved