A336810 Continued fraction expansion of Sum_{k>=0} 1/(k!)!.

2, 1, 1, 179, 2, 1196852626800230399, 1, 1, 179, 1, 1

Offset: 0

Author

Daniel Hoyt, Nov 20 2020

Keywords

• nonn
• cofr

Comments

a(11), a(21), and a(41) have 152, 1349, and 12981 digits, respectively.

Links

• Georg Fischer, Table of n, a(n) for n = 0..20
• Georg Fischer, Table of n, a(n) for n = 0..139
• Daniel Hoyt, Python program that generates the continued fraction from formula.
• Alfred J. van der Poorten and Jeffrey Shallit, Folded continued fractions, Journal of Number Theory, Vol. 40, Issue 2, 1992, pp. 237-250 (cf. prop. 2).

Crossrefs

Cf. A336686 (decimal expansion).

Cf. A001511, A157196, A363841.

Programs

Mathematica

ContinuedFraction[Sum[1/(k!)!, {k, 0, 6}], 21] (* Amiram Eldar, Nov 22 2020 *)

PARI

contfrac(suminf(k=0, 1/(k!)!))

Formula

The peak terms have the form ((k+1)!)! / ((k!)!)^2 - 1. - Georg Fischer, Oct 19 2022 [pers. comm. with J. Shallit]

Let P(k) = ((k+1)!)! / ((k!)!)^2 - 1. After the first term, the rest of the sequence is an interleaving between the n-th runs of '1, 1' and '2' in A157196, and P(A001511(n)+1). - Daniel Hoyt, Jun 26 2023