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 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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. 319-337. [See A317208 for a link.] LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 Aviezri S. Fraenkel, From Enmity to Amity, American Mathematical Monthly 117 (2010) 646-648. Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8. Wolfdieter 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. 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). Row lengths of A189921. Sequence in context: A216647 A072645 A316714 * A360485 A122060 A088939 Adjacent sequences: A135814 A135815 A135816 * A135818 A135819 A135820 KEYWORD nonn,base,easy AUTHOR Wolfdieter Lang, Jan 21 2008 STATUS approved

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Last modified May 26 07:52 EDT 2024. Contains 372807 sequences. (Running on oeis4.)