A016631 Decimal expansion of log(8).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

a(n+1) is also the sequence of digits in the base-ten expansion of the number representing the probability that an acute triangle could be formed with the pieces obtained by breaking a stick into three parts at random. The breaking points are chosen with uniform distribution and independently of one another. - Eugen J. Ionascu, Feb 19 2011

Examples

			2.079441541679835928251696364374529704226500403080765762362040028480180....
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
• Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Eugen J. Ionascu and Gabriel Prajitura, Things to do with a broken stick, arXiv:1009.0890 [math.HO], 2010-2013.
• Index entries for transcendental numbers

Crossrefs

Cf. A016736 (continued fraction). - Harry J. Smith, May 16 2009

Programs

Maple

a:=proc(n)
  local x,y,z,w;
    Digits:=2*n+1;
     x:=3*ln(2);y:=floor(10^(n-2)*x)*10;
       z:=floor(10^(n-1)*x);w:=z-y;
end: # Eugen J. Ionascu, Feb 19 2011

Mathematica

RealDigits[Log[8], 10, 90][[1]] (* Bruno Berselli, Mar 26 2013 *)

PARI

default(realprecision, 20080); x=log(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016631.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

Formula

Equals 2 + Sum_{n >= 1} 1/( n*(16*n^2 - 1) ). This summation was the first problem submitted by Ramanujan to the Journal of the Indian Mathematical Society. See Berndt, Corollary on p. 29. - Peter Bala, Feb 25 2015

Equals 2 + Sum_{n >= 1} (-1)^n*(n-1)/(n*(n+1)). - Bruno Berselli, Sep 09 2020

Equals 2 + Sum_{k>=1} zeta(2*k+1)/16^k. - Amiram Eldar, May 27 2021

Equals 3*A002162. - R. J. Mathar, Apr 11 2024