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A243340
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Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.
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3
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1, 1, 1, 2, 8, 3, 5, 7, 8, 8, 8, 9, 8, 7, 6, 4, 2, 4, 8, 3, 7, 5, 2, 3, 9, 6, 4, 3, 7, 3, 2, 0, 6, 2, 4, 1, 1, 9, 9, 1, 9, 9, 0, 6, 8, 4, 6, 5, 3, 7, 9, 6, 0, 0, 3, 2, 6, 6, 4, 3, 6, 4, 9, 3, 4, 7, 1, 5, 7, 5, 9, 9, 0, 2, 7, 9, 3, 6, 8, 5, 4, 9, 1, 5, 9, 5, 8, 8, 2, 1, 3, 8, 0, 1, 7, 0, 0, 4, 3, 2, 1, 7, 2, 0, 9
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OFFSET
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1,4
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.
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LINKS
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FORMULA
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Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0.
For example, taking k = 0 and k = 1 yields
4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and
4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))).
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EXAMPLE
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1.11283578889876424837523964373206241199199...
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MATHEMATICA
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L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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