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Engel expansion of sqrt(Pi) = 1.77245... .
1

%I #25 Apr 13 2018 11:20:44

%S 1,2,2,12,13,90,121,3457,7372,42530,147081,348753,1185480,17262610,

%T 28408922,87175278,120293961,344697955,634418862,3889219736,

%U 8965899278,9695659687,18962955420,27004920068,67070581522

%N Engel expansion of sqrt(Pi) = 1.77245... .

%C Cf. A006784 for definition of Engel expansion.

%D F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

%H G. C. Greubel, <a href="/A059187/b059187.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%t EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@

%t NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];

%t EngelExp[N[Sqrt[Pi], 7!], 100] (* Modified by _G. C. Greubel_, Dec 27 2016 *)

%Y Cf. A002161 (sqrt(Pi)).

%K nonn,easy,nice

%O 1,2

%A _Mitch Harris_