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A336810
Continued fraction expansion of Sum_{k>=0} 1/(k!)!.
4
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
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).
Sequence in context: A159767 A169658 A330199 * A178473 A164810 A322392
KEYWORD
nonn,cofr
AUTHOR
Daniel Hoyt, Nov 20 2020
STATUS
approved