%I A006784 M4475
%S A006784 1,1,1,8,8,17,19,300,1991,2492,7236,10586,34588,63403,70637,1236467,
%T A006784 5417668,5515697,5633167,7458122,9637848,9805775,41840855,58408380,
%U A006784 213130873,424342175,2366457522,4109464489,21846713216,27803071890
%N A006784 Engel expansion of Pi.
%C A006784 Definition of Pierce expansion : for a real number x (0<x<1), there is 
               always a unique increasing positive integer sequence (a(i))_i>0 such 
               that x = 1/a(1) - 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) -1/a(1)/a(2)/a(3)/
               a(4) .. This expansion can be computed as follows : let u(0)=x and 
               u(k+1)=u(k)/(u(k)-floor(u(k)) then a(n)=floor(u(n)). - Benoit Cloitre, 
               Mar 14 2004
%D A006784 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006784 P. Deheuvels, L'encadrement asymptotique des elements de la serie d'Engel 
               d'un nombre reel, C. R. Acad. Sci. Paris, 295 (1982), 21-24.
%D A006784 F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 
               52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 
               1913, pp. 190-191.
%D A006784 P. Erdos and J. O. Shallit, New bounds on the length of finite Pierce 
               and Engel series. Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 
               1, 43-53.
%D A006784 A. Renyi, A new approach to the theory of Engel's series, Ann. Univ. 
               Sci. Budapest. Eotvos Sect. Math., 5 (1962), 25-32.
%H A006784 S. Plouffe, <a href="b006784.txt">Table of n, a(n) for n = 1..300</a> 
               [There is a limit of about 1000 digits on the size of numbers in 
               b-files]
%H A006784 P. Liardet and P. Stambul, <a href="http://www.emis.de/journals/JTNB/
               2000-1/jtnb12-1_english.html#jourelec">Series d'Engel et fractions 
               continue</a>
%H A006784 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EngelExpansion.html">Engel Expansion</a>
%H A006784 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Pi.html">Pi</a>
%H A006784 <a href="Sindx_El.html#Engel">Index entries for sequences related to 
               Engel expansions</a>
%F A006784 Definition of Engel expansion: For a positive real number x (here Pi), 
               define 1 <= a(1) <= a(2) <= a(3) <= .. so that x = 1/a(1) + 1/a(1)a(2) 
               + 1/a(1)a(2)a(3) + .. by x(1)=x, a(n) = ceil(1/x(n)), x(n+1) = x(n)a(n)-1. 
               Expansion always exists and is unique. See references for more information.
%t A006784 EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ]], First@Transpose@NestList[ 
               {Ceiling[ 1/Expand[ #[[ 1 ]]#[[ 2 ]]-1 ]], Expand[ #[[ 1 ]]#[[ 2 
               ]]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ]]
%t A006784 EngelExp[ N[ Pi, 500000], 27]
%Y A006784 Sequence in context: A022091 A145909 A135405 this_sequence A061156 A109049 
               A160239
%Y A006784 Adjacent sequences: A006781 A006782 A006783 this_sequence A006785 A006786 
               A006787
%K A006784 nonn,nice,easy
%O A006784 1,4
%A A006784 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006784 More terms from Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 10 
               2001