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A135817
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Length of Wythoff representation of n.
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16
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1, 1, 2, 3, 2, 4, 3, 3, 5, 4, 4, 4, 3, 6, 5, 5, 5, 4, 5, 4, 4, 7, 6, 6, 6, 5, 6, 5, 5, 6, 5, 5, 5, 4, 8, 7, 7, 7, 6, 7, 6, 6, 7, 6, 6, 6, 5, 7, 6, 6, 6, 5, 6, 5, 5, 9, 8, 8, 8, 7, 8, 7, 7, 8, 7, 7, 7, 6, 8, 7, 7, 7, 6, 7, 6, 6, 8, 7, 7, 7, 6, 7, 6, 6, 7, 6, 6, 6, 5, 10, 9, 9, 9, 8, 9, 8, 8, 9, 8, 8, 8, 7, 9, 8, 8
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| For the Wythoff representation of n see the W. Lang reference and A189921.
The Wythoff complementary sequences are A(n):=A000201(n) and B(n)=A001950(n), n>=1. The Wythoff representation of n=1 is A(1) and for n>=2 there is a unique representation as composition of A- or B-sequence applied to B(1)=2. E.g. n=4 is A(A(B(1))), written as AAB or as `110`, i.e. 1 for A and 0 for B.
The Wythoff orbit of 1 (starting always with B(1), applying any number of A- or B-sequences) produces every number n>1 just once. This produces a binary Wythoff code for n>1, ending always in 0 (for B(1)). See the W. Lang link for this code.
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REFERENCES
| W. Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp.319-337.
Clark Kimberling, "The Zeckendorf Array Equals the Wythoff Array," The Fibonacci Quarterly 33 (February, 1995) 3-8.
Aviezri S. Fraenkel, "From Enmity to Amity," American Mathematical Monthly 117 (2010) 646-648.
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LINKS
| W. Lang, Wythoff representations for n=1...150.
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FORMULA
| a(n) = number of digits in Wythoff representation of n>=1.
a(n) = length of Wythoff code for n>=1.
a(n) = number of applications of Wythoff sequences A or B on 1 in the Wythoff representation for n >=1.
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EXAMPLE
| W(4) = `110`, i.e. 4 = A(A(B(1))) with Wythoff's A and B sequences.
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CROSSREFS
| Cf. A135818 (number of 1's or A's in Wythoff representation of n).
Cf. A007895 (number of 0's or B's in Wythoff representation of n).
Sequence in context: A205782 A070296 A072645 * A122060 A088939 A004596
Adjacent sequences: A135814 A135815 A135816 * A135818 A135819 A135820
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008, Feb 22 2008, May 21 2008, Sep 08 2008
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