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%I #39 Sep 28 2022 14:08:10
%S 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,
%T 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,
%U 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
%N Decimal expansion of 3*sqrt(3)/(2*Pi).
%C Limiting ratio of areas in the disk-covering problem.
%C From _Daniel Forgues_, May 26 2010: (Start)
%C 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:
%C 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)
%C The area of a regular hexagon circumscribed in a unit-area circle. - _Amiram Eldar_, Nov 05 2020
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Sections 5.9 p. 325 and 8.2 p. 486.
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
%H Veikko Nevanlinna, <a href="https://www.acadsci.fi/mathematica/1973/year1973.html">On constants connected with the prime number theorem for arithmetic progressions</a>, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Product_{n>=1} (1 - 1/(3n)^2). - _Bruno Berselli_, Apr 02 2013
%F Equals sinc(Pi/3). - _Peter Luschny_, Oct 04 2019
%F Equals Product{k>=1} cos(Pi/(3*2^k)). - _Amiram Eldar_, Aug 20 2020
%F Equals Sum_{k>=0} mu(3*k+1)/(3*k+1) (Nevanlinna, 1973). - _Amiram Eldar_, Dec 21 2020
%e 0.8269933431326880742669897474694541620960797205499609791990...
%t 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 *)
%o (PARI) 3*sqrt(3)/(2*Pi) \\ _Michel Marcus_, Nov 05 2020
%Y Cf. A008683, A060544.
%K nonn,cons,easy
%O 0,1
%A _Eric W. Weisstein_, Jul 08 2003
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