A158468 Decimal expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

1, 3, 3, 2, 7, 4, 7, 3, 8, 2, 4, 3, 2, 8, 9, 9, 2, 2, 5, 0, 0, 8, 6, 0, 1, 0, 9, 8, 3, 7, 3, 8, 9, 9, 7, 0, 4, 4, 1, 6, 7, 4, 3, 9, 8, 2, 2, 5, 9, 8, 4, 4, 5, 3, 6, 5, 7, 9, 7, 1, 8, 4, 9, 3, 9, 9, 3, 3, 4, 1, 6, 8, 8, 2, 7, 3, 5, 4, 7, 4, 5, 4, 0, 7, 0, 2, 8, 0, 6, 5, 1, 7, 1, 6, 6, 6, 0, 4, 7, 8, 7, 0, 4, 0, 6, 6, 8, 5

Offset: 1

Author

Alois P. Heinz, Mar 19 2009

Keywords

• nonn
• cons

Comments

Curiously, this constant is close to gamma/log(2)+1/2 = 1.332746177... - Jean-François Alcover, Mar 24 2014

Examples

			1.3327473824328992250086010983738997044167439822598445365797...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A100668 (gamma/log(2)), A158469 (continued fraction), A159835 (Engel expansion), A339168.

Programs

Maple

hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity):
hzs:= convert(evalf(hz/10, 130), string):
seq(parse(hzs[n+1]), n=1..120);

Mathematica

digits = 105; Clear[f]; f[k_] := f[k] = 1 + k - Sum[Exp[-2^j], {j, -k, k}] // RealDigits[#, 10, digits+1]& // First // Quiet; f[1]; f[n=2]; While[f[n] != f[n-1], n++] ; f[n] // Most (* Jean-François Alcover, Feb 19 2013 *)

Formula

Equals gamma/log(2)+1/2 + Sum_{k>=1} Im(Gamma(1-2*k*Pi*i/log(2)))/(k*Pi). - Toshitaka Suzuki, Feb 10 2017

Also equals limit_{k->oo} 1 + Sum_{j>=1} 1-(1-1/2^j)^(2^k). - Toshitaka Suzuki, Feb 12 2017