A135818 Number of 1's (or A's) in the Wythoff representation of n.

1, 0, 1, 2, 0, 3, 1, 1, 4, 2, 2, 2, 0, 5, 3, 3, 3, 1, 3, 1, 1, 6, 4, 4, 4, 2, 4, 2, 2, 4, 2, 2, 2, 0, 7, 5, 5, 5, 3, 5, 3, 3, 5, 3, 3, 3, 1, 5, 3, 3, 3, 1, 3, 1, 1, 8, 6, 6, 6, 4, 6, 4, 4, 6, 4, 4, 4, 2, 6, 4, 4, 4, 2, 4, 2, 2, 6, 4, 4, 4, 2, 4, 2, 2, 4, 2, 2, 2, 0, 9, 7, 7, 7, 5, 7, 5, 5, 7, 5, 5, 5, 3, 7, 5, 5

Offset

1,4

Comments

a(n) = number of applications of Wythoff's A sequence A000201 needed in the unique Wythoff representation of n>=1.

See A135817 for references and links for the Wythoff representation for n>=1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

6 = A(A(A(B(1)))) = AAAB = `1110`, hence a(6)=3.

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; a[n_] := Total[w[n]]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

Crossrefs

Cf. A000201, A135817 (lengths of Wythoff representation), A007895 (number of 0's (or B's) in the Wythoff representation).

Sequence in context: A070812 A308230 A061865 * A078804 A071465 A333409

Adjacent sequences: A135815 A135816 A135817 * A135819 A135820 A135821

Keyword

nonn,base,easy

Author

Wolfdieter Lang, Jan 21 2008

Status

approved