OFFSET
0,1
COMMENTS
Other closed form evaluations of R(p/q):
R(1/4) = Pi/2 - 4/3,
R(1/3) = 1 - log(2),
R(1/2) = 2 - Pi/2,
R(2/3) = 4 - Pi/sqrt(2),
R(1) = log(2),
R(3/2) = Pi + sqrt(3)*log(2 - sqrt(3)),
R(3) = Pi/sqrt(3) - log(2).
LINKS
D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics, page 13.
Jonathan M. Borwein, Ramanujan's Arithmetic-Geometric Mean Continued Fractions and Dynamics
FORMULA
Equivalent formulas:
sqrt(2)*(Pi/2 - log(1 + sqrt(2))),
(Pi - 2*arccoth(sqrt(2)))/sqrt(2),
Integral_{x >= 0} sech(Pi*x/4)/(1 + x^2) dx,
2*Integral_{x = 0..1} sqrt(x)/(1 + x^2) dx,
Integral_{x >= 0} exp(-x/2)*sech(x) dx,
4*Sum_{k >= 1} (-1)^(k+1)/(4*k - 1),
1/2*(-psi(3/8) + psi(7/8)), where psi is the digamma function,
4/3 * 2F1(3/4, 1, 7/4, -1), where 2F1 is the hypergeometric function,
(H(-1/8) - H(-5/8))/2, where H(n) is the n-th harmonic number.
General formula:
The Borwein's closed form formula for R(n) with n integer simplifies to:
R(n) = Pi/2*sec(Pi/(2n)) - 2*sum( cos((k*(n+1)*Pi)/(2*n))*log(2*sin((k*Pi)/(4*n))), {k, 1, 2n-1, 2} ).
Equals 4*A181049. - Peter Bala, Apr 02 2024
EXAMPLE
0.97499098879872209671990033452921084400592...
MATHEMATICA
RealDigits[Sqrt[2]*(Pi/2 - Log[1 + Sqrt[2]]), 10, 100] // First
PROG
(PARI) (psi(7/8)-psi(3/8))/2 \\ Charles R Greathouse IV, Mar 03 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Feb 14 2014
STATUS
approved