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

1,2

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

(4*Pi^2/27) / 10 = 2*Pi^2/135 is the probability that the unique circle that passes through three points that are independently and uniformly selected at random in the interior of an equilateral triangle is entirely contained in that triangle (cf. A395213). - Amiram Eldar, Apr 16 2026

References

Stephen Hawking, God Created the Integers: The Mathematical Breakthroughs That Changed History, Running Press, 2007. See p. 398.

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.

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

Equals A248897^2 = 2*A346933 = A376911/2. - Hugo Pfoertner, Apr 16 2026

Equals Sum_{n>=0} 1/floor((3n+2)/2)^2 [Euler, 1735] (see Hawking). - Stefano Spezia, Jul 16 2026

Example

1.4621636149762012768643690370186...

Maple

evalf(4*Pi^2/27) ;

Mathematica

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

Prog

(PARI) 4*Pi^2/27 \\ G. C. Greubel, Mar 08 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
(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

Crossrefs

Cf. A001651, A011655, A100044, A248897, A346933, A376911, A395213.

Sequence in context: A373635 A160327 A124259 * A154892 A286155 A195860

Adjacent sequences: A214546 A214547 A214548 * A214550 A214551 A214552

Keyword

nonn,cons,easy,changed

Author

R. J. Mathar, Jul 20 2012

Status

approved