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%I #18 May 27 2021 06:24:17
%S 1,4,4,4,6,2,3,9,6,2,4,6,1,6,0,8,1,5,8,8,2,4,9,9,0,9,0,5,2,5,4,8,3,2,
%T 0,3,8,1,3,6,4,2,0,7,1,9,7,8,3,0,7,7,9,1,4,9,5,8,4,3,5,0,7,4,6,0,7,4,
%U 3,5,2,9,6,4,1,5,4,6,7,3,2,0,8,1,8,1,3
%N Decimal expansion of trace of Gaussian operator.
%D Marius Iosifescu, and Cor Kraaikamp, Metrical Theory of Continued Fractions, Springer, 2002, p. 134.
%H Hervé Daudé, Philippe Flajolet and Brigitte Vallée, <a href="https://doi.org/10.1007/3-540-58691-1_52">An analysis of the Gaussian algorithm for lattice reduction</a>, in: L. M. Adleman and M. D. Huang (eds.), Algorithmic Number Theory, First International Symposium, ANTS-I Ithaca, NY, USA, May 6-9, 1994, Proceedings, Lecture Notes in Computer Science, Vol. 877, Springer, Berlin, Heidelberg, 1994, pp. 144-158; <a href="https://hal.inria.fr/inria-00074428/">Inria preprint</a>.
%H Hervé Daudé, Philippe Flajolet and Brigitte Vallée, <a href="https://doi.org/10.1017/S0963548397003258">An average-case analysis of the Gaussian algorithm for lattice reduction</a>, Combinatorics, Probability and computing, Vol. 6, No. 4 (1997), pp. 397-433; <a href="https://hal.archives-ouvertes.fr/inria-00073892/">Inria preprint</a>.
%F From _Amiram Eldar_, May 27 2021: (Start)
%F Equals Sum_{m>=1} 1/(tau(m)^4 + tau(m)^2), where tau(m) = (m + sqrt(m^2+4))/2.
%F Equals 7/2 - 7/(2*sqrt(2)) - 2/sqrt(5) - (1/2) * Sum_{k>=2} (-1)^k * binomial(2*k,k)*(zeta(2*k) - 1 - 1/2^(2*k))*(k - 1)/(k + 1). (End)
%e 0.14446239624616081588249909052548320381...
%t RealDigits[7/2 - 7/(2 Sqrt[2]) - 2/Sqrt[5] + NSum[(-1)^k * Binomial[2*k, k]*(Zeta[2*k] - 1 - 1/2^(2*k))*(k - 1)/(k + 1), {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 500]/2, 10, 100][[1]] (* _Amiram Eldar_, May 27 2021 *)
%K nonn,cons
%O 0,2
%A _N. J. A. Sloane_, Sep 15 2002
%E Offset corrected and more terms added by _Amiram Eldar_, May 27 2021
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