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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).

%I A102364 #20 Sep 29 2018 18:44:01
%S A102364 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,
%T A102364 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,
%U A102364 7,6,6,6,5,6,5,5,7,6,6,6,5,6,5,5,6,5,5
%N 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).
%C A102364 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.
%C A102364 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
%D A102364 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.
%H A102364 Alois P. Heinz, <a href="/A102364/b102364.txt">Table of n, a(n) for n = 0..10946</a>
%H A102364 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibGen.html">General Fibonacci Series</a>
%p A102364 F:= combinat[fibonacci]:
%p A102364 b:= proc(n) option remember; local j;
%p A102364       if n=0 then 0
%p A102364     else for j from 2 while F(j+1)<=n do od;
%p A102364          b(n-F(j))+2^(j-2)
%p A102364       fi
%p A102364     end:
%p A102364 a:= proc(n) local c,m;
%p A102364       c, m:= 0, b(n);
%p A102364       while m>0 do c:= c +1 -irem(m, 2, 'm');
%p A102364       od; c
%p A102364     end:
%p A102364 seq(a(n), n=0..150);  # _Alois P. Heinz_, May 18 2012
%t A102364 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_ *)
%o A102364 (Haskell)
%o A102364 a102364 0 = 0
%o A102364 a102364 n = length $ filter (== 0) $ a213676_row n
%o A102364 -- _Reinhard Zumkeller_, Mar 10 2013
%Y A102364 Cf. A007895, A072649.
%K A102364 nonn
%O A102364 0,4
%A A102364 _Casey Mongoven_, Feb 22 2005