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Decimal expansion of log(4/(1 + sqrt(2))).
1

%I #13 Sep 08 2022 08:45:42

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

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

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

%N Decimal expansion of log(4/(1 + sqrt(2))).

%C Equals Sum_{n>=2, n even} binomial(2n,n)/(n*4^n) = A016627-A091648.

%H G. C. Greubel, <a href="/A157700/b157700.txt">Table of n, a(n) for n = 0..10000</a>

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the Central Binomial Coefficient</a>, Am. Math. Monthly 92, no 7 (1985) 449-457.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.5049207741003475936018...

%p log(4/(1+sqrt(2))) ;

%t RealDigits[Log[4/(1+Sqrt[2])],10,120][[1]] (* _Harvey P. Dale_, Jun 08 2014 *)

%o (PARI) default(realprecision, 100); log(4/(1+sqrt(2))) \\ _G. C. Greubel_, Oct 02 2018

%o (Magma) SetDefaultRealField(RealField(100)); Log(4/(1+Sqrt(2))); // _G. C. Greubel_, Oct 02 2018

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Mar 04 2009