A335846 Decimal expansion of lim_{n->infinity} (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/(2n-1)! - 1/(2n)!))).

5, 4, 8, 7, 7, 3, 5, 4, 7, 0, 4, 0, 8, 5, 6, 8, 7, 5, 1, 3, 0, 6, 9, 9, 2, 2, 7, 4, 0, 6, 9, 1, 4, 5, 5, 5, 6, 2, 6, 0, 0, 0, 4, 6, 7, 3, 8, 0, 3, 0, 4, 6, 4, 8, 9, 7, 2, 6, 3, 9, 1, 4, 7, 2, 0, 2, 0, 5, 1, 1, 1, 5, 1, 6, 7, 9, 5, 2, 4, 9, 1, 9, 0, 1, 6, 3, 1, 3, 8, 6, 1, 0, 3, 6, 4, 7, 6, 1, 8, 9, 4, 1, 3, 3

Offset

0,1

Comments

The sequence of real values x(n) = (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/n! - 1/(n+1)!))) converges to two different limits depending on whether n is even or odd. This integer sequence gives the decimal expansion of the lower limit, to which the odd-indexed terms of {x(n)} converge.

Links

Table of n, a(n) for n=0..103.

Rafik Zeraoulia, Does this a_n = ... have a finite limit?, Math Stackexchange.

Example

0.54877354704085687513069922740691455562600046738030...

Mathematica

(* note that FullSimplify[1/Factorial[i]-1/Factorial[i+1]] == i/Gamma[2 + i]
   which is i/Factorial[1 + i] for integer i *)
sequence = Table[Fold[#2^#1 &, Table[i/(i + 1)!, {i, n, 1, -1}]], {n, 1, 15}];
ListLinePlot[N /@ sequence, PlotRange -> {0, 1}]
N[sequence[[-1]]]
N[sequence[[-2]]]

Prog

(PARI) my(N=99, y=(N/(N+1)!)); forstep(n=N-1, 1, -1, y = ((n/(n+1)!)^y)); y \\ Michel Marcus, Jul 05 2020

Crossrefs

Cf. A328941.

Sequence in context: A334401 A295162 A002340 * A023845 A051555 A309543

Adjacent sequences: A335843 A335844 A335845 * A335847 A335848 A335849

Keyword

nonn,cons

Author

R Zeraoulia, Jun 26 2020

Status

approved