A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)).

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

Offset

0,1

Comments

The formulas for this number and the constant in A113476 are exactly the same except for one small, crucial detail: the infinite sum has a denominator of 3i + 2 rather than 3i + 1, while in the closed form, log(2)/3 is subtracted from rather than added to (Pi * sqrt(3))/9.

Understandably, the typesetter for Spiegel et al. (2009) set the closed formula for this number incorrectly (as being the same as for A113476, compare equation 21.16 on the same page of that book).

References

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (80), page 16.

J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 132.

Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill, 2009, p. 135, equation 21.18.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..2015

Eric W. Weisstein, Euler's Series Transformation.

Formula

Equals Sum_{k >= 0} (-1)^k/(3k + 2) = 1/2 - 1/5 + 1/8 - 1/11 + 1/14 - 1/17 + ... (see A016789).

From Peter Bala, Feb 20 2015: (Start)

Equals (1/2) * Integral_{x = 0..1} 1/(1 + x^(3/2)) dx.

Generalized continued fraction: 1/(2 + 2^2/(3 + 5^2/(3 + 8^2/(3 + 11^2/(3 + ... ))))) due to Euler. For a sketch proof see A024396. (End)

Equals (Psi(5/6)-Psi(1/3))/6. - Vaclav Kotesovec, Jun 16 2015

Equals Integral_{x = 1..infinity} 1/(1 + x^3) dx. - Robert FERREOL, Dec 23 2016

Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 2) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A008544(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 2)). - Peter Bala, Dec 01 2021

From Bernard Schott, Jan 28 2022: (Start)

Equals Integral_{x = 0..1} x/(1+ x^3) dx (see Rivaud reference).

Equals 3 * A196548. (End)

From Peter Bala, Mar 03 2024: (Start)

Equals (1/2)*hypergeom([2/3, 1], [5/3], -1).

Gauss's continued fraction: 1/(2 + 2^2/(5 + 3^2/(8 + 5^2/(11 + 6^2/(14 + 8^2/(17 + 9^2/(20 + 11^2/(23 + 12^2/(26 + ... ))))))))). (End)

Example

0.373550727891424180392282045394659721402855371244161773816401641964909853052219...

Maple

evalf((Psi(5/6)-Psi(1/3))/6, 120); # Vaclav Kotesovec, Jun 16 2015

Mathematica

RealDigits[(Pi Sqrt[3])/9 - (Log[2]/3), 10, 100][[1]]

Prog

(PARI) (Pi/sqrt(3)-log(2))/3 \\ Charles R Greathouse IV, Jul 29 2011
(PARI)
default(realprecision, 98);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(3*n+2)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

Crossrefs

Cf. A073010, A193535, A024396, A113476, A196548, A258969.

Sequence in context: A247217 A252734 A101636 * A096247 A337013 A362026

Adjacent sequences: A193531 A193532 A193533 * A193535 A193536 A193537

Keyword

nonn,cons

Author

Alonso del Arte, Jul 29 2011

Status

approved