A222466 - OEIS

%I #13 Aug 16 2017 13:41:37

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

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

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

%N Decimal expansion of the limit of the continued fraction 1/(1+2/(2+2/(3+2/(4+... in terms of Bessel functions.

%C The continued fraction (0 + K_{k=1}^{infinity}(2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... converges, and its limit is given in the formula section in terms of Bessel functions.

%C In general, the continued fraction 0 + K_{k=1}^{infinity}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/ Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel functions. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems. This results in the given formula 0 + K_{k=1}^{infinity}(x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).

%C For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

%H G. C. Greubel, <a href="/A222466/b222466.txt">Table of n, a(n) for n = 0..5000</a>

%F (0 + K_{k=1}^{infinity}(2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... =

%F sqrt(2)*BesselI(1,2*sqrt(2))/(2*BesselI(0,2*sqrt(2)))

%F =  0.5631786198117113854257529037035635...

%t RealDigits[BesselI[1, 2*Sqrt[2]]/(Sqrt[2]*BesselI[0, 2*Sqrt[2]]), 10, 50][[1]] (* _G. C. Greubel_, Aug 16 2017 *)

%o (PARI)

%o default(realprecision, 120);

%o sqrt(2)*besseli(1,2*sqrt(2))/(2*besseli(0,2*sqrt(2))) \\ _Rick L. Shepherd_, Jan 18 2014

%Y A052119 (x=1), 2*A221913/A084950.

%K nonn,cons

%O 0,1

%A _Wolfdieter Lang_, Mar 07 2013

%E Offset corrected and terms added by _Rick L. Shepherd_, Jan 18 2014