OFFSET
1,1
COMMENTS
If u(0) = exp(1/m) with m an 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..501 [a(1)=7 inserted by Georg Fischer, Nov 20 2020]
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.
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Pelegrí Viader, Lluís Bibiloni, and 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->oo} 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) default(realprecision, 100000); r=exp(2); for(n=1, 100, s=(r/(r-floor(r))); print1(floor(r), ", "); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Mar 09 2004
EXTENSIONS
a(1)=7 inserted by Georg Fischer, Nov 20 2020
STATUS
approved