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A245780
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Decimal expansion of (1-C_2)/e, a constant connected with two-sided generalized Fibonacci sequences, where C_2 is the Euler-Gompertz constant.
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0
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1, 4, 8, 4, 9, 5, 5, 0, 6, 7, 7, 5, 9, 2, 2, 0, 4, 7, 9, 1, 8, 3, 5, 9, 9, 9, 4, 7, 0, 1, 3, 3, 9, 2, 1, 8, 4, 1, 4, 7, 6, 3, 8, 3, 7, 6, 2, 4, 8, 5, 9, 6, 2, 6, 9, 2, 9, 8, 5, 8, 1, 8, 8, 6, 2, 3, 8, 9, 2, 7, 9, 7, 1, 8, 5, 7, 5, 8, 2, 5, 8, 6, 3, 4, 9, 3, 7, 0, 2, 3, 3, 1, 0, 7, 8, 2, 3, 9, 3, 7, 9
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OFFSET
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0,2
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.
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LINKS
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Walther Janous, Problem 1552, Crux Mathematicorum, Vol. 16, No. 6 (1990), p. 171; Solution to Problem 1552, by Richard Katz, ibid., Vol. 17, No. 7 (1991), pp. 223-224.
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FORMULA
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Equals 1/e + Ei(-1), where Ei is the exponential integral function.
Equals Integral_{x=0..1} exp(-1/x) dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=1..+oo} exp(-x)/x^2 dx. - Jianing Song, Oct 03 2021
Equals lim_{n->oo} (Sum_{k=1..n-1} (k/(k+1))^n)/n (Janous, 1990). - Amiram Eldar, Apr 03 2022
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EXAMPLE
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0.148495506775922047918359994701339218414763837624859626929858...
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MATHEMATICA
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$RecursionLimit = 10^4; digits = 101; m0 = 100; dm = 100; Clear[g]; g[m_] := g[m] = (Clear[a, b, f]; b[n_] := 2*n; a[n_ /; n >= m] = 0; a[1] = 1; a[2] = -1; a[n_] := -(n-1)^2; f[m] = b[m]; f[n_] := f[n] = b[n] + a[n+1]/f[n+1]; (1 - f[0])/E); g[m0]; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits] != RealDigits[g[m - dm], 10, digits], m = m + dm]; RealDigits[g[m], 10, digits] // First
(* or, as verification: *) RealDigits[1/E + ExpIntegralEi[-1], 10, digits] // First
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PROG
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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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