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A091832 Pierce expansion of 1/e^2. 1
18, 19, 136, 349, 357, 1354, 6996, 7135, 9531, 11558, 15996, 17432, 52118, 151048, 427802, 821834, 877819, 972918, 1046690, 1540789, 3653077, 8200738, 9628573, 164153335, 5607624822, 86457467082, 141885251873, 151882622551 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If u(0)=exp(1/m) m integer>=1 and u(n+1)=u(n)/frac(u(n)) then floor(u(n))=m*n

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .

Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.

Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a problem of Alfred Renyi, Economics Working Paper No. 340.

FORMULA

let u(0)=exp(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n))

1/e^2 = 1/a(1) - 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) - 1/a(1)/a(2)/a(3)/a(4)...

limit n-> infty a(n)^(1/n)=e

MATHEMATICA

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/E^2, 7!], 15] (* G. C. Greubel, Nov 14 2016 *)

PROG

(PARI) r=exp(2); for(n=1, 10, r=r/(r-floor(r)); print1(floor(r), ", "))

CROSSREFS

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A059194.

Sequence in context: A041670 A041672 A041674 * A041676 A041678 A197352

Adjacent sequences:  A091829 A091830 A091831 * A091833 A091834 A091835

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Mar 09 2004

STATUS

approved

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Last modified September 20 01:58 EDT 2019. Contains 327207 sequences. (Running on oeis4.)