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A069887
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Number of terms in the simple continued fraction expansion for (1+1/n)^n.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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Asymptotically it seems that a(n) ~ C*n*log(n) where C = 0.84... is close to the constant described in A055573.
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EXAMPLE
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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.
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MATHEMATICA
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Table[Length[ContinuedFraction[(1+1/n)^n]], {n, 70}] (* Harvey P. Dale, Jun 12 2013 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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