A135818 - OEIS

%I #12 Jul 01 2023 09:19:47

%S 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,

%T 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,

%U 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

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

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

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

%H Amiram Eldar, <a href="/A135818/b135818.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

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

%K nonn,base,easy

%O 1,4

%A _Wolfdieter Lang_, Jan 21 2008