A317208 The Wythoff representation of n: an alternative way of presenting A189921.

0, 1, 2, 12, 112, 22, 1112, 212, 122, 11112, 2112, 1212, 1122, 222, 111112, 21112, 12112, 11212, 2212, 11122, 2122, 1222, 1111112, 211112, 121112, 112112, 22112, 111212, 21212, 12212, 111122, 21122, 12122, 11222, 2222, 11111112, 2111112, 1211112, 1121112

Offset

0,3

Comments

This is an encoding of the position of n in the A000201, A001950 "Wythoff" table T.

Let T denote the following 3-rowed table, whose rows are n, A = A000201(n), B = A001950(n):

n: 1 2 3 .4 .5 .6 .7 .8 .9 ...

A: 1 3 4 .6 .8 .9 11 12 14 ...

B: 2 5 7 10 13 15 18 20 23 ...

Set a(0)=0. For n>0, locate n in rows A and B of the table, and indicate how to reach that entry starting from column 1. For example, 18 = B(7) = B(B(3)) = B(B(A(2))) = B(B(A(B(1)))), so the path to reach 18 is BBAB, which we write (encoding A as 1, B as 2) as a(18) = 2212.

This is another way of writing the Wythoff representation of n described in Lang (1996) and A189921.

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.

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000

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. [Corrected scanned copy, with permission of the author.]

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]; a[n_] := FromDigits[ReplaceAll[w[n], {0 :> 2}]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 01 2023 *)

Crossrefs

Cf. A189921, A135817 (length).

Cf. also A317207.

Cf. A000201, A001950.

Sequence in context: A264916 A296644 A235860 * A207778 A102659 A212659

Adjacent sequences: A317205 A317206 A317207 * A317209 A317210 A317211

Keyword

nonn,base

Author

N. J. A. Sloane, Aug 09 2018

Extensions

a(23) and beyond from Lars Blomberg, Aug 11 2018

Status

approved