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A006784 Engel expansion of Pi.
(Formerly M4475)
109
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 (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 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
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
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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Olivier Gérard, Jul 10 2001
STATUS
approved

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Last modified June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)