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A073010 Decimal expansion of Pi/sqrt(27). 45
6, 0, 4, 5, 9, 9, 7, 8, 8, 0, 7, 8, 0, 7, 2, 6, 1, 6, 8, 6, 4, 6, 9, 2, 7, 5, 2, 5, 4, 7, 3, 8, 5, 2, 4, 4, 0, 9, 4, 6, 8, 8, 7, 4, 9, 3, 6, 4, 2, 4, 6, 8, 5, 8, 5, 2, 3, 2, 9, 4, 9, 7, 8, 4, 6, 2, 7, 0, 7, 7, 2, 7, 0, 4, 2, 1, 1, 7, 9, 6, 1, 2, 2, 8, 0, 4, 1, 6, 6, 2, 7, 3, 7, 3, 5, 3, 3, 8, 9, 6, 1, 8, 7, 4, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Original name: Decimal expansion of sum(1/(n*binomial(2*n,n)), n=1..infinity).
This appears to be Pi/sqrt(27). See A111510. - Marco Matosic, Feb 27 2008
This is Pi*sqrt(3)/9 = A019676*A002194, see eq. (12) in Lehmer link. - R. J. Mathar, Mar 04 2009
Value of the Dirichlet L-series of the non-principal character modulo m=3 (A102283) at s=1. - R. J. Mathar, Oct 03 2011
Construct the largest possible circle inside a given equilateral triangle. This constant is the ratio of the area of the circle to the area of the triangle (A245670 is analogous square in triangle). - Rick L. Shepherd, Jul 29 2014
REFERENCES
L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (81), page 16.
LINKS
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005), 25-36.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019. See r(q) for q=3 in Table 1, p. 7.
D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457. See eq. (12).
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991), 53-55.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (1.6.2)
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
FORMULA
-Pi/(3*sqrt(3)) = Sum_{n>=0} (1/(6*n+1) - 2/(6*n+2) - 3/(6*n+3) - 1/(6*n+4) + 2/(6*n+5) + 3/(6*n+6)). - Mats Granvik, Sep 08 2013
Equals Integral_{0..oo} 2*x/((x^2+1)*(x^4+x^2+1)) dx. - Jean-François Alcover, Sep 10 2013
From Peter Bala, Feb 16 2015: (Start)
Pi/sqrt(27) = Sum_{n >= 0} 1/((3*n + 1)*(3*n + 2)) = 1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ....
Continued fraction: 1/(1 + 1^2/(1 + 2^2/(2 + 4^2/(1 + 5^2/(2 + ... + (3*n + 1)^2/(1 + (3*n + 2)^2/(2 + ... ))))))).
Pi/sqrt(27) = Integral_{t = 0..1/2} 1/(t^2 - t + 1) dt = Integral_{t = 0..1/2} (1 + t - t^3 - t^4)/(1 - t^6) dt.
Pi/sqrt(27) = (1/4)*Sum_{n >= 0} (-1/8)^n * (9*n + 5)/((3*n + 1)*(3*n + 2)).
BBP-type formulas:
Pi/sqrt(27) = Sum_{n >= 0} (1/64)^(n+1)*( 32/(6*n + 1) + 16/(6*n + 2) - 4/(6*n + 4) - 2/(6*n + 5) ) follows from the above integral representation.
Pi/sqrt(27) = Sum_{n >= 0} (-1)^n*(1/27)^(n+1)*( 9/(6*n + 1) + 9/(6*n + 2) + 6/(6*n + 3) + 3/(6*n + 4) + 1/(6*n + 5) ) follows from the result: Pi/3 = Integral_{t = 0..1/sqrt(3)} 1/(1 - sqrt(3)*t + t^2) dt. (End)
Equals Integral_{x=0..oo} x*I_0(x)*K_0(x)^2 dx over a triple product of modified Bessel functions. - R. J. Mathar, Oct 14 2015
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(exp(x) + exp(-x) + 1) dx.
Equals Integral_{x=0..oo} 1/(1 + x + x^2 + x^3 + x^4 + x^5) dx. (End)
Equals (3*S - 4)/8, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Product_{p prime} (1 - Kronecker(-3, p)/p)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p)^(-1). - Amiram Eldar, Nov 06 2023
From Peter Bala, Dec 09 2023: (Start)
Pi/sqrt(27) = Sum_{n >= 1} 1/(n*binomial(2*n,n)) = (1/6)*Sum_{n >= 1} 3^n/(n*binomial(2*n,n)) (see Lehmer, equation 12, and also p. 456).
Pi/sqrt(27) = (1/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n,n)).
Pi/sqrt(27) = (9/4)*Sum_{n >= 3} (n - 1)*(n - 2)/binomial(2*n,n). (End)
Equals integral_{x=0..oo} 1/(1-x^3) dx [Nahin]. - R. J. Mathar, May 16 2024
EXAMPLE
0.60459978807807261686469275254738524409468...
MATHEMATICA
RealDigits[ N [Sum[1/(n*Binomial[2n, n]), {n, 1, Infinity}], 110]] [[1]]
RealDigits[Pi/(3*Sqrt[3]), 10, 105][[1]] (* T. D. Noe, Sep 11 2013 *)
PROG
(PARI) Pi/sqrt(27) \\ Charles R Greathouse IV, Sep 11 2013
(Magma) R:=RealField(106); SetDefaultRealField(R); n:=Pi(R)/Sqrt(27); Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 12 2018
CROSSREFS
Sequence in context: A298528 A341325 A021947 * A100120 A132709 A197148
KEYWORD
nonn,cons,easy,changed
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved

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Last modified May 25 06:40 EDT 2024. Contains 372782 sequences. (Running on oeis4.)