A069880 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k!.

1, 2, 3, 5, 6, 9, 9, 13, 14, 18, 19, 20, 24, 24, 23, 29, 33, 36, 31, 38, 41, 42, 46, 50, 53, 58, 56, 57, 70, 73, 77, 69, 76, 76, 78, 77, 80, 85, 89, 101, 101, 105, 106, 104, 106, 112, 115, 124, 113, 126, 124, 124, 130, 144, 144, 148, 140, 149, 141, 151, 157, 158, 172

Offset: 1

Author

Benoit Cloitre, May 04 2002

Keywords

• easy
• nonn

Examples

			For n=4, Sum_{k=1..n} 1/k! = 1/1! + 1/2! + 1/3! + 1/4! = 1/1 + 1/2 + 1/6 + 1/24 = 41/24 = 1 + 1/(1 + 1/(2 + 1/(2 + 1/3))) = CF[1;1,2,2,1], so a(4) = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Amiram Eldar, Plot of a(n)/(n * log(log(n))) for n = 1..10000.

Crossrefs

Cf. A055573, A061354, A061355, A070267.

Programs

Mathematica

lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k!, {k, 1, 100}]] (* Amiram Eldar, Apr 30 2022 *)

Formula

Does lim_{n->infinity} a(n)/(n * log(log(n))) = C = 2.XXX...?