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Length of Wythoff representation of n.
17

%I #37 Jul 02 2023 06:59:40

%S 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,

%T 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,

%U 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

%N Length of Wythoff representation of n.

%C For the Wythoff representation of n see the W. Lang reference and A189921.

%C 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.

%C 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.

%D 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.]

%H Amiram Eldar, <a href="/A135817/b135817.txt">Table of n, a(n) for n = 1..10000</a>

%H Aviezri S. Fraenkel, <a href="http://www.jstor.org/stable/10.4169/000298910x496787">From Enmity to Amity</a>, American Mathematical Monthly 117 (2010) 646-648.

%H Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly 33 (1995) 3-8.

%H Wolfdieter Lang, <a href="/A135817/a135817.pdf">Wythoff representations for n=1...150</a>.

%F a(n) = number of digits in Wythoff representation of n>=1.

%F a(n) = length of Wythoff code for n>=1.

%F a(n) = number of applications of Wythoff sequences A or B on 1 in the Wythoff representation for n >=1.

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

%t 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 *)

%Y Cf. A135818 (number of 1's or A's in Wythoff representation of n).

%Y Cf. A007895 (number of 0's or B's in Wythoff representation of n).

%Y Row lengths of A189921.

%K nonn,base,easy

%O 1,3

%A _Wolfdieter Lang_, Jan 21 2008