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

%I A159835 #40 Feb 16 2025 08:33:10
%S A159835 1,4,4,4,4,6,11,11,11,14,61,266,1006,1030,1261,6264,7583,7979,7986,
%T A159835 12386,80041,87434,130927,270073,1653819,1715177,1973657,3483485,
%U A159835 12346987,17531499,21237674,84103203,195088616,725688944,2813572082,3138084145,10870485195
%N A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).
%C A159835 Cf. A006784 for definition of Engel expansion.
%D A159835 F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
%H A159835 F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A159835 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A159835 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A159835 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%H A159835 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%e A159835 hz = 1.3327473824328992250086010983738997044167439822598445365797 ...
%p A159835 hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity):
%p A159835 engel:= (r,n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
%p A159835 Digits:=300:
%p A159835 engel(evalf(hz), 39);
%Y A159835 Cf. A158468 (decimal expansion), A158469 (continued fraction).
%K A159835 easy,nice,nonn
%O A159835 1,2
%A A159835 _Alois P. Heinz_, Apr 23 2009
%E A159835 Some terms corrected by _Alois P. Heinz_, Nov 22 2020