OFFSET
0,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.
LINKS
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.
Peter Fishburn, Andrew Odlyzko and Fred Roberts, Two-sided generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 27, No. 4 (1989), pp. 352-361.
FORMULA
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
EXAMPLE
0.148495506775922047918359994701339218414763837624859626929858...
MATHEMATICA
$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
PROG
(PARI) 1/exp(1) - eint1(1, 1)[1] \\ Michel Marcus, Aug 06 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Aug 01 2014
STATUS
approved