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A321943
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Decimal expansion of Ni_1 = (1/2)*(gamma - log(2*Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).
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1
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3, 6, 9, 6, 6, 9, 2, 9, 9, 2, 4, 6, 0, 9, 3, 6, 8, 8, 5, 2, 2, 9, 2, 6, 3, 0, 8, 6, 3, 5, 5, 8, 3, 5, 7, 5, 6, 5, 9, 6, 8, 2, 1, 9, 4, 3, 3, 2, 1, 7, 8, 3, 8, 6, 5, 8, 5, 7, 3, 2, 0, 7, 6, 9, 5, 9, 6, 6, 8, 1, 6, 7, 4, 6, 1, 5, 7, 1, 9, 3, 7, 7, 7, 3, 7, 3, 0
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OFFSET
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0,1
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COMMENTS
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This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).
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REFERENCES
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D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.
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LINKS
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FORMULA
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Ni_1 = Sum_{k>=2} (-1)^k*zeta(k)/(k+1).
Ni_1 = Sum_{n>0} (Integral_{x=0..1} x^2*(1-x)_{n-1} dx)/(n*n!), where (z)_n = z*(z+1)*(z+2)*...*(z+n-1) is the Pochhammer symbol.
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EXAMPLE
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0.369669299246093688522926308635583575659682194332178386585...
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MAPLE
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Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);
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MATHEMATICA
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First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]
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PROG
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(PARI) (1/2)*(Euler-log(2*Pi))+1
(Python)
from mpmath import *
mp.dps = 100; mp.pretty = True
+(1/2)*(euler-log(2*pi))+1
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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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