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%I #11 Feb 27 2021 21:54:07
%S 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,
%T 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,
%U 1,9,7,0,9,4,6,6,4,4,2,8,9,5,0,6,9,0,8
%N Decimal expansion of the probability of two consecutive continued fraction coefficients being both even, when the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.
%H V. N. Nolte, <a href="https://doi.org/10.1016/0019-3577(90)90025-I">Some probabilistic results on the convergents of continued fractions</a>, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.
%F 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))).
%e 0.1169400035780680765605607509208534105...
%o (PARI)
%o 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)
%o (PARI)
%o C = log(2)-1+(log(72*Pi)-4*log(gamma(1/4)))/log(2)
%o 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)
%Y Cf. A340533, A340543.
%K nonn,cons
%O 0,3
%A _A.H.M. Smeets_, Feb 16 2021
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