A159835 -id:A159835 - cp's OEIS frontend

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

Alois P. Heinz, Mar 19 2009

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

			1.3327473824328992250086010983738997044167439822598445365797...

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

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

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);

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 *)

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

A158469 Continued fraction for hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

Original entry on oeis.org

1, 3, 189, 3, 2, 2, 1, 5, 4, 1, 1, 3, 1, 1, 1, 5, 8, 12, 1, 22, 7, 14, 1, 2, 1, 5, 1, 4, 222, 1, 1, 2, 3, 24, 6, 27, 1, 15, 1, 9, 1, 1, 18, 6, 24, 2, 1, 7, 1, 4, 2, 2, 1, 1, 84, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 15, 3, 13, 3, 2, 14, 1, 1, 1, 10, 15, 10, 1, 6, 120, 1, 31, 2, 4, 2, 7, 2, 2, 1, 1, 1, 1, 1, 3, 7

Offset: 0

Alois P. Heinz, Mar 19 2009

			1.33274738243289922500860109837389970441674398225984453657972 ...

Cf. A158468 (decimal expansion), A159835 (Engel expansion).

with(numtheory): hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity): cfrac(evalf(hz, 130), 100, 'quotients')[];

terms = 95; digits = terms+15; 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++]; hz = FromDigits[f[n]]*10^-digits; ContinuedFraction[hz, terms] (* Jean-François Alcover, Mar 23 2017 *)

cp's OEIS Frontend

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

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