A158469 -id:A158469 - 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

A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

Original entry on oeis.org

1, 4, 4, 4, 4, 6, 11, 11, 11, 14, 61, 266, 1006, 1030, 1261, 6264, 7583, 7979, 7986, 12386, 80041, 87434, 130927, 270073, 1653819, 1715177, 1973657, 3483485, 12346987, 17531499, 21237674, 84103203, 195088616, 725688944, 2813572082, 3138084145, 10870485195

Offset: 1

Alois P. Heinz, Apr 23 2009

Cf. A006784 for definition of Engel expansion.

			hz = 1.3327473824328992250086010983738997044167439822598445365797 ...

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A158468 (decimal expansion), A158469 (continued fraction).

hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity):
engel:= (r,n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=300:
engel(evalf(hz), 39);

Some terms corrected by Alois P. Heinz, Nov 22 2020

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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