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A006784
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Engel expansion of Pi.
(Formerly M4475)
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95
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1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, 7236, 10586, 34588, 63403, 70637, 1236467, 5417668, 5515697, 5633167, 7458122, 9637848, 9805775, 41840855, 58408380, 213130873, 424342175, 2366457522, 4109464489, 21846713216, 27803071890
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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
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REFERENCES
| 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.
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
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.
A. Renyi, A new approach to the theory of Engel's series, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 5 (1962), 25-32.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| S. Plouffe, Table of n, a(n) for n = 1..300 [There is a limit of about 1000 digits on the size of numbers in b-files]
P. Liardet and P. Stambul, Series d'Engel et fractions continue
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Pi
Index entries for sequences related to Engel expansions
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FORMULA
| 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.
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MATHEMATICA
| 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 ]]
EngelExp[ N[ Pi, 500000], 27]
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CROSSREFS
| Sequence in context: A145909 A168409 A135405 * A168456 A061156 A195862
Adjacent sequences: A006781 A006782 A006783 * A006785 A006786 A006787
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms from Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 10 2001
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