A336810 Continued fraction expansion of Sum_{k>=0} 1/(k!)!.
2, 1, 1, 179, 2, 1196852626800230399, 1, 1, 179, 1, 1
Offset: 0
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).
Programs
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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
Comments