

A135817


Length of Wythoff representation of n.


16



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
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OFFSET

1,3


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 Bsequence 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 Bsequences) 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.


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.319337.


LINKS

Table of n, a(n) for n=1..105.
Aviezri S. Fraenkel, From Enmity to Amity, American Mathematical Monthly 117 (2010) 646648.
C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 38.
W. Lang, Wythoff representations for n=1...150.


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.


EXAMPLE

W(4) = `110`, i.e. 4 = A(A(B(1))) with Wythoff's A and B sequences.


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: A070296 A216647 A072645 * A122060 A088939 A004596
Adjacent sequences: A135814 A135815 A135816 * A135818 A135819 A135820


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang Jan 21 2008, Feb 22 2008, May 21 2008, Sep 08 2008


STATUS

approved



