This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135817 #39 Jan 05 2025 19:51:38
%S A135817 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 A135817 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 A135817 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 A135817 Length of Wythoff representation of n.
%C A135817 For the Wythoff representation of n see the W. Lang reference and A189921.
%C A135817 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 A135817 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 A135817 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 A135817 Amiram Eldar, <a href="/A135817/b135817.txt">Table of n, a(n) for n = 1..10000</a>
%H A135817 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 A135817 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly 33 (1995) 3-8.
%H A135817 Wolfdieter Lang, <a href="/A135817/a135817.pdf">Wythoff representations for n=1...150</a>.
%F A135817 a(n) = number of digits in Wythoff representation of n>=1.
%F A135817 a(n) = length of Wythoff code for n>=1.
%F A135817 a(n) = number of applications of Wythoff sequences A or B on 1 in the Wythoff representation for n >=1.
%e A135817 W(4) = `110`, i.e., 4 = A(A(B(1))) with Wythoff's A and B sequences.
%t A135817 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 A135817 Cf. A135818 (number of 1's or A's in Wythoff representation of n).
%Y A135817 Cf. A007895 (number of 0's or B's in Wythoff representation of n).
%Y A135817 Row lengths of A189921.
%K A135817 nonn,base,easy
%O A135817 1,3
%A A135817 _Wolfdieter Lang_, Jan 21 2008