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A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)). 5
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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

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

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

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, A258969.

Sequence in context: A247217 A252734 A101636 * A096247 A122583 A001265

Adjacent sequences:  A193531 A193532 A193533 * A193535 A193536 A193537

KEYWORD

nonn,cons

AUTHOR

Alonso del Arte, Jul 29 2011

STATUS

approved

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Last modified March 24 06:56 EDT 2019. Contains 321444 sequences. (Running on oeis4.)