

A113476


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


8



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
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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: McGrawHill (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. 204216
L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5
Index entries for transcendental numbers


FORMULA

Equals int_{0}^{1}dx/(1+x^3) = sum(k>=0, (1)^k/(3k+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


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. A073010, A193535, A024217, A193534.
Sequence in context: A196654 A019728 A265183 * A171043 A124599 A005601
Adjacent sequences: A113473 A113474 A113475 * A113477 A113478 A113479


KEYWORD

cons,nonn


AUTHOR

Benoit Cloitre, Jan 08 2006


STATUS

approved



