This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A222471 #21 Feb 23 2025 04:11:15
%S A222471 1,4,3,9,7,4,9,3,2,1,8,7,0,2,3,2,8,0,5,8,9,5,7,0,6,9,5,7,4,1,1,2,2,7,
%T A222471 4,2,5,1,5,2,7,3,3,7,6,2,2,3,8,1,1,6,1,7,5,2,8,1,4,5,3,0,7,8,8,7,7,2,
%U A222471 3,6,1,6,8,1,6,4,3,4,5,9,6,3,8,5,0,1,9,5,1,3,1,8,5,9,7,7,0,4,8,7,6,3,4,1,7,8,7,4,0,2
%N A222471 Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.
%C A222471 The continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4- ... converges, and its negative limit is given in the formula section in terms of Bessel functions.
%C A222471 In general, the continued fraction 0 + K_{k=1..oo} (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 function. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems for given x. This results in the formula 0 + K_{k=1..oo} (x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).
%C A222471 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 A222471 G. C. Greubel, <a href="/A222471/b222471.txt">Table of n, a(n) for n = 1..5000</a>
%F A222471 Equals (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)).
%e A222471 -1.4397493218702328058...
%t A222471 RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* _G. C. Greubel_, Aug 16 2017 *)
%o A222471 (PARI) besselj(1,sqrt(8))/besselj(0,sqrt(8))/sqrt(2) \\ _Charles R Greathouse IV_, Feb 19 2014
%Y A222471 Cf. A052119 (x=1), A222466 (x=2), A222469/A222470.
%K A222471 nonn,cons
%O A222471 1,2
%A A222471 _Wolfdieter Lang_, Mar 23 2013