A006784 Engel expansion of Pi. (Formerly M4475)

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, 31804388758, 32651669133

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 éléments de la série d'Engel d'un nombre réel, 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.

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..711

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.

P. Erdős 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.

P. Liardet and P. Stambul, Séries d'Engel et fractions continuées, Journal de Théorie des Nombres de Bordeaux 12 (2000), 37-68.

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) = ceiling(1/x(n)), x(n+1) = x(n)a(n)-1. Expansion always exists and is unique. See references for more information.

Example

1/1 + 1/1 + 1/1 + 1/8 + 1/(8*8) + 1/(8*8*17) <= Pi < 1/1 + 1/1 + 1/1 + 1/8 + 1/(8*8) + 1/(8*8*16), so a(6) = 17. - Peter Munn, Aug 14 2022

Maple

a(n):=proc(s)
local
i, j, max, aa, bb, lll, prod, S, T, kk;
    S := evalf(abs(s));
    max := 10^(Digits - 10);
    prod := 1;
    lll := [];
    while prod <= max do
        T := 1 + trunc(1/S);
        S := frac(S*T);
        lll := [op(lll), T];
        prod := prod*T
    end do;
    RETURN(lll)
end: # Simon Plouffe, Apr 24 2016

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 A346532

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