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A074904
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Decimal expansion of trace of Gaussian operator.
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0
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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, 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, 3, 5, 2, 9, 6, 4, 1, 5, 4, 6, 7, 3, 2, 0, 8, 1, 8, 1, 3
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OFFSET
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0,2
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REFERENCES
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Marius Iosifescu, and Cor Kraaikamp, Metrical Theory of Continued Fractions, Springer, 2002, p. 134.
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LINKS
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Hervé Daudé, Philippe Flajolet and Brigitte Vallée, An analysis of the Gaussian algorithm for lattice reduction, 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; Inria preprint.
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FORMULA
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Equals Sum_{m>=1} 1/(tau(m)^4 + tau(m)^2), where tau(m) = (m + sqrt(m^2+4))/2.
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)
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EXAMPLE
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0.14446239624616081588249909052548320381...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Offset corrected and more terms added by Amiram Eldar, May 27 2021
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STATUS
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approved
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