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A102364
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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).
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10
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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
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a102364 0 = 0
a102364 n = length $ filter (== 0) $ a213676_row n
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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