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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_{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 Cf. A002162: R(1) = log(2); A180434: R(1/2) = 2-Pi/2. Sequence in context: A361603 A232735 A191760 * A109846 A096230 A114433 Adjacent sequences: A237838 A237839 A237840 * A237842 A237843 A237844 KEYWORD nonn,cons AUTHOR Jean-François Alcover, Feb 14 2014 STATUS approved

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Last modified August 5 02:16 EDT 2024. Contains 374935 sequences. (Running on oeis4.)