A245780 Decimal expansion of (1-C_2)/e, a constant connected with two-sided generalized Fibonacci sequences, where C_2 is the Euler-Gompertz constant.

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

Offset: 0

Author

Jean-François Alcover, Aug 01 2014

Keywords

• nonn
• cons

Examples

			0.148495506775922047918359994701339218414763837624859626929858...
		

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.

Crossrefs

Cf. A073003 (C_2), A099285 (C_2 / e).

Programs

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

PARI

1/exp(1) - eint1(1,1)[1] \\ Michel Marcus, Aug 06 2020

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