A301484 Decimal expansion of J_0(2)/J_1(2) = 1 - 1/(2 - 1/(3 - 1/(4 - ...))).

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

Offset: 0

Author

David H Bailey, Mar 22 2018

Keywords

• nonn
• cons

Comments

These are the first 105 decimal digits of the constant defined by the continued fraction 1 - 1/(2 - 1/(3 - 1/(4 - ... -1/m))) as m goes to infinity. The continued fraction appears to converge fairly rapidly. Just 50 terms, for instance, suffices to produce a numerical value that appears to be good to 100 digits, based on comparisons with more terms and higher precision. This sequence was brought to the author's attention by Beresford Parlett of U.C. Berkeley.

Addendum: This sequence has been identified by Karl Dilcher. He noted that the sequence of continued fraction convergents is the same as A058797. In short, the real constant whose decimal expansion is given above is given by BesselJ[0,2]/BesselJ[1,2] (Mathematica expression). The comments at A058797 have quite a bit of additional information and references.

Equivalent to the imaginary part of the infinite continued fraction i + 1/(2i + 1/(3i + ...)) where i = sqrt(-1). - Matthew Niemiro, Dec 22 2019

Examples

			0.38821076556779578751165855730653702922174504072532981864642827593735174...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A058797, A091681, A296168.

Programs

Magma

SetDefaultRealField(RealField(100)); BesselFunction(0, 2)/BesselFunction(1, 2); // G. C. Greubel, Dec 31 2019

Maple

evalf(BesselJ(0, 2)/BesselJ(1, 2), 100); # G. C. Greubel, Dec 31 2019

Mathematica

1 +ContinuedFractionK[(-1)^(n+1)*n, {n,2,Infinity}]
N[1+ContinuedFractionK[(-1)^(n+1)*n, {n,2,50}], 105] (* 105 decimals *)
RealDigits[BesselJ[0, 2]/BesselJ[1, 2], 10, 100][[1]] (* G. C. Greubel, Dec 31 2019 *)

PARI

default(realprecision, 100); besselj(0,2)/besselj(1,2) \\ Altug Alkan, Mar 22 2018

Sage

numerical_approx(bessel_J(0,2)/bessel_J(1,2), digits=100) # G. C. Greubel, Dec 31 2019

Formula

Equals BesselJ(0,2)/BesselJ(1,2) = A091681 / A348607.