A356749 a(n) is the number of trailing 1's in the dual Zeckendorf representation of n (A104326).

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

Offset

0,4

Comments

The asymptotic density of the occurrences of k = 0, 1, 2, ... is 1/phi^(k+2), where phi = 1.618033... (A001622) is the golden ratio.

The asymptotic mean of this sequence is phi.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Example

  n  a(n)  A104326(n)
  -  ----  ----------
  0     0           0
  1     1           1
  2     0          10
  3     2          11
  4     1         101
  5     0         110
  6     3         111
  7     0        1010
  8     2        1011
  9     1        1101

Mathematica

fb[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; f[v_] := Module[{m = Length[v], k}, k = m; While[v[[k]] == 1, k--]; m - k]; a[n_] := Module[{v = fb[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i ;; i + 2]] == {1, 0, 0}, v[[i ;; i + 2]] = {0, 1, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, f[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

Crossrefs

Cf. A001622, A104326.

Similar sequences: A003849, A035614, A276084, A278045.

Sequence in context: A143153 A127448 A128179 * A178780 A058558 A210869

Adjacent sequences: A356746 A356747 A356748 * A356750 A356751 A356752

Keyword

nonn,base

Author

Amiram Eldar, Aug 25 2022

Status

approved