A091832 Pierce expansion of 1/e^2.

7, 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

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.

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, 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

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A059194.

Sequence in context: A185455 A103570 A223247 * A374460 A256011 A272972

Adjacent sequences: A091829 A091830 A091831 * A091833 A091834 A091835

Keyword

nonn

Author

Benoit Cloitre, Mar 09 2004

Extensions

a(1)=7 inserted by Georg Fischer, Nov 20 2020

Status

approved