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A237841 Decimal expansion of Ramanujan's AGM Continued Fraction R(2) = R_1(2,2). 0
9, 7, 4, 9, 9, 0, 9, 8, 8, 7, 9, 8, 7, 2, 2, 0, 9, 6, 7, 1, 9, 9, 0, 0, 3, 3, 4, 5, 2, 9, 2, 1, 0, 8, 4, 4, 0, 0, 5, 9, 2, 0, 2, 1, 9, 9, 9, 4, 7, 1, 0, 6, 0, 5, 7, 4, 5, 2, 6, 8, 2, 5, 1, 2, 8, 5, 8, 7, 7, 3, 8, 7, 4, 5, 5, 7, 0, 8, 5, 9, 4, 3, 5, 2, 3, 2, 5, 3, 2, 0, 9, 1, 1, 1, 2, 9, 3, 6, 2, 5 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..99.

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(sech(Pi*x/4)/(1 + x^2), {x, 0, infinity}),

2*integral(sqrt(x)/(1 + x^2), {x, 0, 1}),

integral(exp(-x/2)*sech(x), {x, 0, infinity}),

4*sum((-1)^(k + 1)/(4*k - 1), {k, 1, infinity}),

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)/(2n))*log(2*sin((k*Pi)/(4n))), {k, 1, 2n-1, 2} ).

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

Cf. A002162: R(1) = log(2); A180434: R(1/2) = 2-Pi/2.

Sequence in context: A010546 A232735 A191760 * A109846 A096230 A114433

Adjacent sequences:  A237838 A237839 A237840 * A237842 A237843 A237844

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Feb 14 2014

STATUS

approved

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Last modified November 19 06:34 EST 2019. Contains 329310 sequences. (Running on oeis4.)