OFFSET
1
COMMENTS
The row lengths sequence of this array is A135817.
For the Wythoff representation of n see the W. Lang reference.
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` or here as 1,1,0, 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)).
REFERENCES
Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) 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..9415 (first 1000 rows)
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (February, 1995) 3-8.
Wolfdieter Lang, Wythoff representations for n=1...150.
FORMULA
The entries of row n, n>=1, are given by W(n), computed with the algorithm given on p. 335 of the W. Lang reference. 1 is used for Wythoff's A sequence and 0 for the B sequence.
EXAMPLE
n=1: 1;
n=2: 0;
n=3: 1, 0;
n=4: 1, 1, 0;
n=5: 0, 0;
n=6: 1, 1, 1, 0;
n=7: 0, 1, 0;
n=8: 1, 0, 0;
...
1 = A(1); 2 = B(1), 3 = A(B(1)), 4 = A(A(B(1))),
5 = B(B(1)), 6 = A(A(A(B(1)))), 7 = B(A(B(1))),
8 = A(B(B(1))), ...
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; Table[w[n], {n, 1, 25}] // Flatten (* Amiram Eldar, Jul 01 2023 *)
CROSSREFS
KEYWORD
nonn,easy,tabf,base
AUTHOR
Wolfdieter Lang, Jun 12 2011
STATUS
approved