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A135817
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Length of Wythoff representation of n.
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17
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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
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OFFSET
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1,3
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COMMENTS
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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
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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. 319-337. [See A317208 for a link.]
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LINKS
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FORMULA
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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
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W(4) = `110`, i.e., 4 = A(A(B(1))) with Wythoff's A and B sequences.
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MATHEMATICA
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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 *)
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CROSSREFS
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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).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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