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A069887 Number of terms in the simple continued fraction expansion for (1+1/n)^n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

lim_{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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Sequence in context: A199590 A096624 A145378 * A254340 A120303 A093413

Adjacent sequences:  A069884 A069885 A069886 * A069888 A069889 A069890

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 04 2002

STATUS

approved

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Last modified February 18 15:08 EST 2018. Contains 299324 sequences. (Running on oeis4.)