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
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
Programs
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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.
Comments