A214549 Decimal expansion of 4*Pi^2/27.

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

Offset: 1

Author

R. J. Mathar, Jul 20 2012

Keywords

• nonn
• cons
• easy

Comments

Represents the value of the Dirichlet series for A011655 (principal Dirichlet character mod 3) at s=2.

Equals the asymptotic mean of the abundancy index of the numbers that are not divisible by 3 (A001651). - Amiram Eldar, May 12 2023

Examples

			1.4621636149762012768643690370186...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• R. J. Mathar, Table of Dirichlet L-Series, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
• Index entries for transcendental numbers.

Crossrefs

Cf. A001651, A011655, A100044.

Programs

Julia

using Nemo
R = RealField(310)
t = const_pi(RR) + const_pi(RR); s = t * t
s / RR(27) |> println # Peter Luschny, Mar 13 2018

Magma

R:= RealField(); 4*Pi(R)^2/27; // G. C. Greubel, Mar 08 2018

Magma

R:=RealField(106); SetDefaultRealField(R); n:=4*Pi(R)^2/27; Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 13 2018

Maple

evalf(4*Pi^2/27) ;

Mathematica

RealDigits[(4Pi^2)/27,10,120][[1]] (* Harvey P. Dale, Dec 20 2012 *)

PARI

4*Pi^2/27 \\ G. C. Greubel, Mar 08 2018

Formula

Equals (4/3)*A100044.

Equals Sum_{n>=0} (1/(3*n+1)^2 + 1/(3*n+2)^2).

From Peter Luschny, May 13 2020: (Start)

Equals (8/9) * Sum_(k>=1) 1/k^2 =8/9 *A013661.

Equals -(16/9) * Sum_(k>=1) (-1)^k/k^2 = -16/9 * A072691.

Equals (64/27) * ( Integral_{x=0..1} sqrt(1 - x^2) )^2 = 64/27 * A091476. (End)

Equals Integral_{x=0..oo} log(x)/(x^3 - 1) dx. - Amiram Eldar, Aug 12 2020