A336810 - OEIS

%I #60 Aug 05 2023 22:15:22

%S 2,1,1,179,2,1196852626800230399,1,1,179,1,1

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

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

%H Georg Fischer, <a href="/A336810/b336810.txt">Table of n, a(n) for n = 0..20</a>

%H Georg Fischer, <a href="/A336810/a336810.txt">Table of n, a(n) for n = 0..139</a>

%H Daniel Hoyt, <a href="/A336810/a336810_2.txt">Python program that generates the continued fraction from formula</a>.

%H Alfred J. van der Poorten and Jeffrey Shallit, <a href="https://doi.org/10.1016/0022-314X(92)90042-N">Folded continued fractions</a>, Journal of Number Theory, Vol. 40, Issue 2, 1992, pp. 237-250 (cf. prop. 2).

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

%F 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

%t ContinuedFraction[Sum[1/(k!)!, {k, 0, 6}], 21] (* _Amiram Eldar_, Nov 22 2020 *)

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

%Y Cf. A336686 (decimal expansion).

%Y Cf. A001511, A157196, A363841.

%K nonn,cofr

%O 0,1

%A _Daniel Hoyt_, Nov 20 2020