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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. 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The continued fraction (0 + K_{k=1}^{infinity}(-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.

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 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 n-th 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/(1-2/(2-2/(3-2/(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

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Last modified October 31 11:28 EDT 2020. Contains 338101 sequences. (Running on oeis4.)