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A244261
Decimal expansion of c = 2.4149..., a random mapping statistics constant such that the asymptotic expectation of the maximum rho length (graph diameter) in a random n-mapping is c*sqrt(n).
2
2, 4, 1, 4, 9, 0, 1, 0, 2, 3, 7, 1, 7, 6, 1, 6, 2, 4, 1, 1
OFFSET
1,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.
P. Flajolet and A. M. Odlyzko, Random Mapping Statistics, Advances in Cryptology - EUROCRYPT '89, J.-J. Quisquater and J. Vandewalle (eds.), Lecture Notes in Computer Science, Springer Verlag, 1990, pp. 329-354.
FORMULA
I(x) = integral_(0..x) (exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))) dy,
c = sqrt(Pi/2)*integral_(0..infinity) 1 - exp(Ei(-x) - I(x)) dx, where Ei is the exponential integral function.
EXAMPLE
2.4149010237176162411...
MAPLE
evalf(sqrt(Pi/2)*Int(1 - exp(Ei(-x) - Int((exp(-y)/y)*(1 - exp(-2*(y/(exp(x - y) - 1)))), y=0..x)), x=0..infinity)); # Vaclav Kotesovec, Aug 12 2019
MATHEMATICA
digits = 20; m0 = 100; dm = 10; I0[x_?NumericQ] := NIntegrate[(Exp[-y]/y)*(1 - Exp[-2*(y/(Exp[x - y] - 1))]), {y, 0, x}, WorkingPrecision -> digits+5]; Clear[f]; f[m_] := f[m] = Sqrt[Pi/2]* NIntegrate[1 - Exp[ExpIntegralEi[-x] - I0[x]], {x, 0, m}, WorkingPrecision -> digits+5]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], Print["m = ", m]; m = m + dm]; RealDigits[f[m], 10, digits] // First
CROSSREFS
KEYWORD
nonn,cons,more
AUTHOR
STATUS
approved