A069887 Number of terms in the simple continued fraction expansion for (1+1/n)^n.

1, 2, 5, 7, 7, 10, 14, 16, 24, 16, 20, 29, 39, 40, 42, 39, 46, 42, 44, 57, 59, 55, 66, 55, 57, 70, 68, 81, 86, 81, 91, 109, 106, 108, 119, 117, 123, 118, 124, 118, 120, 133, 142, 147, 164, 155, 159, 164, 167, 163, 177, 176, 168, 171, 198, 198, 201, 201, 205, 206, 227

Offset: 1

Author

Benoit Cloitre, May 04 2002

Keywords

• easy
• nonn

Comments

Limit_{n -> infinity} (1+1/n)^n = e.

For any natural number N, limit_{n->infinity} (log(N)^(1/n) + 1/n)^n = e*log(N). - Alexander R. Povolotsky, Dec 06 2007

Examples

			The simple continued fraction for (1+1/10)^10 is [2, 1, 1, 2, 5, 1, 128, 1, 2, 12, 5, 3, 46, 1, 11, 7] which contains 16 elements, hence a(10) = 16.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
• Amiram Eldar, Plot of a(n)/(n*log(n)) for n = 2..10000

Crossrefs

Cf. A001113, A055573, A069655.

Programs

Mathematica

Table[Length[ContinuedFraction[(1+1/n)^n]],{n,70}] (* Harvey P. Dale, Jun 12 2013 *)

Formula

Asymptotically it seems that a(n) ~ C*n*log(n) where C = 0.84... is close to the constant described in A055573.