

A222471


Decimal expansion of the negative of the limit of the continued fraction 1/(12/(22/(32/(4... in terms of Bessel functions.


3



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, 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, 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
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OFFSET

1,2


COMMENTS

The continued fraction (0 + K_{k=1}^{infinity}(2/k)/(2) = 1/(12/(22/(32/(4 ... converges, and its negative limit is given in the formula section in terms of Bessel functions.
In general, the continued fraction 0 + K_{k=1}^{infinity}(x/k) = x/(1+x/(2+x/(3+... has nth 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}^{infinity}(x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x).
For x=1 see for the limit of the continued fraction A052119 and for the nth approximation A001053(n+1)/A001040(n+1).


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000


FORMULA

(0 + K_{k=1}^{infinity}(2/k)/(2) = 1/(12/(22/(32/(4 ... =
(1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2))
= 1.4397493218702328058...


MATHEMATICA

RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)


PROG

(PARI) besselj(1, sqrt(8))/besselj(0, sqrt(8))/sqrt(2) \\ Charles R Greathouse IV, Feb 19 2014


CROSSREFS

Cf. A052119 (x=1), A222466 (x=2), A222469/A222470.
Sequence in context: A123596 A309460 A200361 * A180858 A263193 A094885
Adjacent sequences: A222468 A222469 A222470 * A222472 A222473 A222474


KEYWORD

nonn,cons


AUTHOR

Wolfdieter Lang, Mar 23 2013


STATUS

approved



