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

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

Offset

0,1

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A336686 (decimal expansion).

Cf. A001511, A157196, A363841.

Sequence in context: A159767 A169658 A330199 * A178473 A164810 A322392

Adjacent sequences: A336807 A336808 A336809 * A336811 A336812 A336813

Keyword

nonn,cofr

Author

Daniel Hoyt, Nov 20 2020

Status

approved