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

0,4

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

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 *)

Prog

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

Crossrefs

Cf. A007895, A072649.

Sequence in context: A097367 A130211 A317207 * A342774 A132923 A144329

Adjacent sequences: A102361 A102362 A102363 * A102365 A102366 A102367

Keyword

nonn

Author

Casey Mongoven, Feb 22 2005

Status

approved