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A341641 Decimal expansion of the probability of two consecutive continued fraction coefficients being both even, when the continued fraction coefficients satisfy the Gauss-Kuzmin distribution. 0
1, 1, 6, 9, 4, 0, 0, 0, 3, 5, 7, 8, 0, 6, 8, 0, 7, 6, 5, 6, 0, 5, 6, 0, 7, 5, 0, 9, 2, 0, 8, 5, 3, 4, 1, 0, 5, 7, 2, 6, 6, 5, 5, 6, 5, 8, 2, 1, 8, 6, 7, 0, 1, 5, 6, 8, 8, 1, 8, 1, 1, 5, 4, 4, 2, 7, 0, 7, 1, 9, 7, 0, 9, 4, 6, 6, 4, 4, 2, 8, 9, 5, 0, 6, 9, 0, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..86.

V. N. Nolte, Some probabilistic results on the convergents of continued fractions, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.

FORMULA

Equals Sum_{j >= 1} log_2(Gamma(1+1/(4*j+2))/Gamma(1+(j+1)/(2*j+1))*Gamma(1+(2*j+1)/4/j)/Gamma(1+1/4/j))).

EXAMPLE

0.1169400035780680765605607509208534105...

PROG

(PARI)

sumpos(j=1, log(gamma(1+1/(4*j+2))/gamma(1+(j+1)/(2*j+1))*gamma(1+(2*j+1)/4/j)/gamma(1+1/4/j)))/log(2)

(PARI)

C = log(2)-1+(log(72*Pi)-4*log(gamma(1/4)))/log(2)

C+sumpos(n=2, (-1)^n*(zeta(n)-1)/n*((2^(2-n)-2^(2-2*n)-1)*(zeta(n)-1)+(2^(n-1)-1)*2^(2-2*n)))/log(2)

CROSSREFS

Cf. A340533, A340543.

Sequence in context: A085138 A346176 A225053 * A215483 A153872 A242208

Adjacent sequences:  A341638 A341639 A341640 * A341642 A341643 A341644

KEYWORD

nonn,cons

AUTHOR

A.H.M. Smeets, Feb 16 2021

STATUS

approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)