A189921 Wythoff representation of natural numbers.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0

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

See A317208 for another encoding; also for the link to the scanned W. Lang article with corrections.

Cf. A135817 (length), A189920 (Zeckendorf).

Sequence in context: A106344 A106346 A296212 * A341346 A167371 A127241

Adjacent sequences: A189918 A189919 A189920 * A189922 A189923 A189924

Keyword

nonn,easy,tabf,base

Author

Wolfdieter Lang, Jun 12 2011

Status

approved