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

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• base
• easy

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)