cp's OEIS Frontend

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.

%I A245780 #33 Apr 03 2022 13:42:32
%S A245780 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,
%T A245780 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,
%U A245780 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
%N 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.
%D A245780 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.
%H A245780 Walther Janous, <a href="https://cms.math.ca/publications/crux/issue?volume=16&amp;issue=6">Problem 1552</a>, Crux Mathematicorum, Vol. 16, No. 6 (1990), p. 171; <a href="https://cms.math.ca/publications/crux/issue?volume=17&amp;issue=7">Solution to Problem 1552</a>, by Richard Katz, ibid., Vol. 17, No. 7 (1991), pp. 223-224.
%H A245780 Peter Fishburn, Andrew Odlyzko and Fred Roberts, <a href="http://www.mathstat.dal.ca/FQ/Scanned/27-5/fishburn.pdf">Two-sided generalized Fibonacci sequences</a>, The Fibonacci Quarterly, Vol. 27, No. 4 (1989), pp. 352-361.
%F A245780 Equals 1/e + Ei(-1), where Ei is the exponential integral function.
%F A245780 Equals Integral_{x=0..1} exp(-1/x) dx. - _Amiram Eldar_, Aug 06 2020
%F A245780 Equals Integral_{x=1..+oo} exp(-x)/x^2 dx. - _Jianing Song_, Oct 03 2021
%F A245780 Equals lim_{n->oo} (Sum_{k=1..n-1} (k/(k+1))^n)/n (Janous, 1990). - _Amiram Eldar_, Apr 03 2022
%e A245780 0.148495506775922047918359994701339218414763837624859626929858...
%t A245780 $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
%t A245780 (* or, as verification: *) RealDigits[1/E + ExpIntegralEi[-1], 10, digits] // First
%o A245780 (PARI) 1/exp(1) - eint1(1,1)[1] \\ _Michel Marcus_, Aug 06 2020
%Y A245780 Cf. A073003 (C_2), A099285 (C_2 / e).
%K A245780 nonn,cons
%O A245780 0,2
%A A245780 _Jean-François Alcover_, Aug 01 2014