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A140881
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An asymptotic limit sequence of a non-algebraic number limit: Start term:a(1)=Log[4]/Log[3]; a(n) = a(n - 1)^(1 + 1/(n - 1)); Out_n=Floor[a(n)].
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0
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0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 41, 52, 65, 83, 104, 132, 166, 210, 265, 335, 422, 533, 673, 849, 1072, 1353, 1707, 2154, 2718, 3430, 4329, 5462, 6893, 8698, 10975, 13850, 17476, 22053, 27828, 35115, 44310, 55913, 70555, 89030, 112344
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| The asymptotic recursion is derived like this:
Mathematica gives:
Limit[Fibonacci[n]^(1/n), n -> Infinity] ==(1+Sqrt[5])/2
True
( Binet like result for a Fermat of Beta integer type)
The same limit is given by:
Limit[Fibonacci[n + 1]/Fibonacci[n], n -> Infinity]
Asymptotic sequence is in the Limit:
Fibonacci[n]^(1/n)=Fibonacci[n+1]/Fibomacci[n];
or a(n)=a(n-1)^(1+1/(n-1)).
This current sequence gives a low ratio limit
that approaches the transcendental limit of Log[4]/Log[3].
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FORMULA
| Start term:a(1)=Log[4]/Log[3]; a(n) = a(n - 1)^(1 + 1/(n - 1)); Out_n=Floor[a(n)].
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MATHEMATICA
| Clear[a] a[0] = 0; a[1] = Log[4]/Log[3]; a[n_] := a[n] = a[n - 1]^(1 + 1/(n - 1)) w = Table[Floor[a[n]], {n, 0, 50}]
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CROSSREFS
| Cf. A000045.
Sequence in context: A179241 A157046 A017979 * A063827 A127217 A057042
Adjacent sequences: A140878 A140879 A140880 * A140882 A140883 A140884
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KEYWORD
| nonn,uned,tabl
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 22 2008
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