1,4

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

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.

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

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.

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]

Adjacent sequences: A006781 A006782 A006783 this_sequence A006785 A006786 A006787

Sequence in context: A022091 A145909 A135405 this_sequence A061156 A109049 A160239

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms from Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 10 2001