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.
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
Offset: 0
Examples
0.1169400035780680765605607509208534105...
Links
- V. N. Nolte, Some probabilistic results on the convergents of continued fractions, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.
Programs
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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)
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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)
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)).