OFFSET
1,1
COMMENTS
log(8) - 2 is 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
From Amiram Eldar, May 22 2026: (Start)
log(8) - 2 is the probability that when a rod AB is marked at random with points P and Q, and a point R is then taken at random in PQ, then PR^2 + RQ^2 > AP^2 + QB^2 (Wolstenholme, 1871).
log(8) - 2 is the probability that the distances from the sides of a point uniformly selected at random in the interior of an equilateral triangle can be the lengths on an acute triangle (McCall, 1871; Simmons, 1885). (End)
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, 1985.
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].
Frank Budden, Some Connected Problems on Probability, The Mathematical Gazette, Vol. 66, No. 435 (1982), pp. 4-10.
Florian Cajori, Problem 301, with solutions by Artemas Martin and Henry Heaton, Mathematical Messenger, Vol. 5, No. 2 (1888), pp. 173-175.
Eugen J. Ionascu and Gabriel Prajitura, Things to do with a broken stick, International Journal of Geometry, Vol. 2, No. 2 (2013), pp. 5-30; arXiv preprint, arXiv:1009.0890 [math.HO], 2010-2013.
Hugh McCall, Problem 3342, with a solution by Stephen Watson, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 15, 1871, pp. 58-59.
W. J. Miller, Problem 1100, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 2, 1865, pp. 74-75; Solution, by Augustus De Morgan, ibid., Vol. 5, 1866, pp. 33-34.
W. J. Miller, Problem 1100, with a solution by Hugh McColl, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 15, 1871, pp. 35-36.
W. J. Miller, Problem 1100, with a solution by W. T. A. Emtage, F. P. Matz, and others, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 37, 1882, p. 73.
T. C. Simmons, Problem 7849, with a solution by Hugh McCall, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 42, 1885, pp. 105-107.
Joseph Wolstenholme, Problem 3310, with a solution by Stephen Watson, Mathematical Questions with Their Solutions, from the "Educational Times", Vol. 15, 1871, pp. 57-58.
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
From Peter Bala, Nov 09 2025: (Start)
log(8) = 17/8 - Integral_{x = 0..1} x^3/(1 + x)^3 dx
log(8) = 17/8 - (9/2)*Sum_{n >= 1} 1/((2*n)*(2*n + 1)*(2*n + 2)*(2*n + 3)).
log(8) = 17/8 - (3/2)*Sum_{n >= 1} (-1)^(n+1)/((n + 1)*(n + 2)*(n + 3)). (End)
EXAMPLE
2.079441541679835928251696364374529704226500403080765762362040028480180....
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved