A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.

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

Offset

0,1

Comments

This number is transcendental - this follows from a result of Baker (1968) on linear forms of algebraic numbers.

References

Jolley, Summation of Series, Dover (1961), eq (79) 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 21.16

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

A. Baker, Linear forms in the logarithms of algebraic numbers (IV). Mathematika, 15 (1968) pp. 204-216

L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5

Eric W. Weisstein, Euler's Series Transformation.

Index entries for transcendental numbers

Formula

Equals Integral_{x = 0..1} dx/(1+x^3) = Sum_{k >= 0} (-1)^k/(3*k+1) = 1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 + ... (see A016777). - Benoit Cloitre, Alonso del Arte, Jul 29 2011

Generalized continued fraction: 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + 10^2/(3 + ... ))))) due to Euler. For a sketch proof see A024217. - Peter Bala, Feb 22 2015

Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 1) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A007559(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 1)). - Peter Bala, Dec 01 2021

From Peter Bala, Mar 03 2024: (Start)

Equals hypergeom([1/3, 1], [4/3], -1).

Gauss's continued fraction: 1/(1 + 1/(4 + 3^2/(7 + 4^2/(10 + 6^2/(13 + 7^2/(16 + 9^2/(19 + 10^2/(22 + 12^2/(25 + 13^2/(28 + ... )))))))))). (End)

Example

0.835648848264721053337... = A073010 + A193535.

Mathematica

RealDigits[(Log[2]+\[Pi]/Sqrt[3])/3, 10, 120][[1]]  (* Harvey P. Dale, Mar 26 2011 *)

Prog

(PARI) 1/3*(log(2)+Pi/sqrt(3))

Crossrefs

Cf. A007559, A016777, A024217, A073010, A193534, A193535.

Sequence in context: A196654 A019728 A265183 * A171043 A124599 A365959

Adjacent sequences: A113473 A113474 A113475 * A113477 A113478 A113479

Keyword

cons,nonn

Author

Benoit Cloitre, Jan 08 2006

Status

approved