cp's OEIS Frontend

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

%I A193534 #73 Mar 03 2024 09:04:40
%S A193534 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,
%T A193534 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,
%U A193534 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
%N A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)).
%C A193534 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.
%C A193534 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).
%D A193534 L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (80), page 16.
%D A193534 J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 132.
%D A193534 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.
%H A193534 Gheorghe Coserea, <a href="/A193534/b193534.txt">Table of n, a(n) for n = 0..2015</a>
%H A193534 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/EulersSeriesTransformation.html">Euler's Series Transformation</a>.
%F A193534 Equals Sum_{k >= 0} (-1)^k/(3k + 2) = 1/2 - 1/5 + 1/8 - 1/11 + 1/14 - 1/17 + ... (see A016789).
%F A193534 From _Peter Bala_, Feb 20 2015: (Start)
%F A193534 Equals (1/2) * Integral_{x = 0..1} 1/(1 + x^(3/2)) dx.
%F A193534 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)
%F A193534 Equals (Psi(5/6)-Psi(1/3))/6. - _Vaclav Kotesovec_, Jun 16 2015
%F A193534 Equals Integral_{x = 1..infinity} 1/(1 + x^3) dx. - _Robert FERREOL_, Dec 23 2016
%F A193534 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
%F A193534 From _Bernard Schott_, Jan 28 2022: (Start)
%F A193534 Equals Integral_{x = 0..1} x/(1+ x^3) dx (see Rivaud reference).
%F A193534 Equals 3 * A196548. (End)
%F A193534 From _Peter Bala_, Mar 03 2024: (Start)
%F A193534 Equals (1/2)*hypergeom([2/3, 1], [5/3], -1).
%F A193534 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)
%e A193534 0.373550727891424180392282045394659721402855371244161773816401641964909853052219...
%p A193534 evalf((Psi(5/6)-Psi(1/3))/6, 120); # _Vaclav Kotesovec_, Jun 16 2015
%t A193534 RealDigits[(Pi Sqrt[3])/9 - (Log[2]/3), 10, 100][[1]]
%o A193534 (PARI) (Pi/sqrt(3)-log(2))/3 \\ _Charles R Greathouse IV_, Jul 29 2011
%o A193534 (PARI)
%o A193534 default(realprecision, 98);
%o A193534 eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(3*n+2)))), "3..-2")) \\ _Gheorghe Coserea_, Oct 06 2015
%Y A193534 Cf. A073010, A193535, A024396, A113476, A196548, A258969.
%K A193534 nonn,cons
%O A193534 0,1
%A A193534 _Alonso del Arte_, Jul 29 2011