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A006784 Engel expansion of Pi.
(Formerly M4475)
102
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; text; internal format)
OFFSET

1,4

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

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 Jeffrey 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).

LINKS

Simon 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

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.

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]

CROSSREFS

Sequence in context: A145909 A168409 A135405 * A214830 A168456 A232652

Adjacent sequences:  A006781 A006782 A006783 * A006785 A006786 A006787

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Olivier Gérard, Jul 10 2001

STATUS

approved

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Last modified October 25 19:50 EDT 2014. Contains 248557 sequences.