A102364 Number of terms in Fibonacci sequence less than n not used in Zeckendorf representation of n (the Zeckendorf representation of n is a sum of non-consecutive distinct Fibonacci numbers).

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

Offset: 0

Author

Casey Mongoven, Feb 22 2005

Keywords

• nonn

Comments

Number of 0's in Zeckendorf-binary representation of n. For example, the Zeckendorf representation of 12 is 8+3+1, which is 10101 in binary notation.

For n > 0: number of zeros in n-th row of A213676, or, number of zeros in n-th row of A189920. - Reinhard Zumkeller, Mar 10 2013

References

• E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10946
• Ron Knott, General Fibonacci Series

Crossrefs

Cf. A007895, A072649.

Programs

Haskell

a102364 0 = 0
a102364 n = length $ filter (== 0) $ a213676_row n
-- Reinhard Zumkeller, Mar 10 2013

Maple

F:= combinat[fibonacci]:
b:= proc(n) option remember; local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= proc(n) local c,m;
      c, m:= 0, b(n);
      while m>0 do c:= c +1 -irem(m, 2, 'm');
      od; c
    end:
seq(a(n), n=0..150);  # Alois P. Heinz, May 18 2012

Mathematica

F = Fibonacci; b[n_] := b[n] = Module[{j}, If[n==0, 0, For[j=2, F[j+1] <= n, j++]; b[n-F[j]]+2^(j-2)]]; a[n_] := Module[{c, m}, {c, m} = {0, b[n]}; While[m>0, c = c + 1 - Mod[m, 2]; m = Floor[m/2]]; c]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 09 2016, after Alois P. Heinz *)