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

0,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.

Links

Table of n, a(n) for n=0..100.

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

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

Sequence in context: A091198 A200641 A377046 * A165267 A092159 A322258

Adjacent sequences: A245777 A245778 A245779 * A245781 A245782 A245783

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 01 2014

Status

approved