A006784 Engel expansion of Pi.

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice
• easy

Comments

Definition of Pierce expansion: for a real number x (0i>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(1)=1/x and u(k+1) = u(k)/(u(k)-floor(u(k))); then a(n)=floor(u(n)). - _Benoit Cloitre, Mar 14 2004 [corrected by Jason Yuen, Dec 29 2024]

Examples

			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
		

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

Programs

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]

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.

Extensions

More terms from Olivier Gérard, Jul 10 2001