A086089 Decimal expansion of 3*sqrt(3)/(2*Pi).

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

Offset: 0

Author

Eric W. Weisstein, Jul 08 2003

Keywords

• nonn
• cons
• easy

Comments

Limiting ratio of areas in the disk-covering problem.

From Daniel Forgues, May 26 2010: (Start)

Consider: A060544 (Centered 9-gonal numbers), starting with a(1)=1, P_c(9, n), n >= 1. Every third triangular number, starting with a(1)=1, P(3, 3n-2), n >= 1. Then:

1/(Sum_{n=0..infinity} 1/binomial(3n+2,2)) = 1/(Sum_{n=1..infinity} 1/binomial(3n-1,2)) = 1/(Sum_{n=1..infinity} 1/P_c(9,n)) = 1/(Sum_{n=1..infinity} 1/P(3,3n-2)) = 1/(Sum_{n=1..infinity} 1/A060544(n)) = this constant. (End)

The area of a regular hexagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020

Examples

			0.8269933431326880742669897474694541620960797205499609791990...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, Sections 5.9 p. 325 and 8.2 p. 486.
• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.

Links

• Veikko Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
• Index entries for transcendental numbers

Crossrefs

Cf. A008683, A060544.

Programs

Mathematica

RealDigits[3 Sqrt[3]/(2 Pi), 10, 110][[1]] (* or, from the third comment: *) RealDigits[N[Product[1 - 1/(3 n)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PARI

3*sqrt(3)/(2*Pi) \\ Michel Marcus, Nov 05 2020

Formula

Equals Product_{n>=1} (1 - 1/(3n)^2). - Bruno Berselli, Apr 02 2013

Equals sinc(Pi/3). - Peter Luschny, Oct 04 2019

Equals Product{k>=1} cos(Pi/(3*2^k)). - Amiram Eldar, Aug 20 2020

Equals Sum_{k>=0} mu(3*k+1)/(3*k+1) (Nevanlinna, 1973). - Amiram Eldar, Dec 21 2020