

A135817


Length of Wythoff representation of n.


17



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

Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (editors), Application of Fibonacci numbers, vol. 6, Kluwer, Dordrecht, 1996, pp. 319337. [See A317208 for a link.]


LINKS



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.


MATHEMATICA

z[n_] := Floor[(n + 1)*GoldenRatio]  n  1; h[n_] := z[n]  z[n  1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; w[0] = 0; a[n_] := Length[w[n]]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)


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).


KEYWORD

nonn,base,easy


AUTHOR



STATUS

approved



