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A035517
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Triangular array read by rows, formed from Zeckendorf expansion of integers: repeatedly subtract the largest Fibonacci number you can until nothing remains. Row n give Z. expansion of n.
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31
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0, 1, 2, 3, 1, 3, 5, 1, 5, 2, 5, 8, 1, 8, 2, 8, 3, 8, 1, 3, 8, 13, 1, 13, 2, 13, 3, 13, 1, 3, 13, 5, 13, 1, 5, 13, 2, 5, 13, 21, 1, 21, 2, 21, 3, 21, 1, 3, 21, 5, 21, 1, 5, 21, 2, 5, 21, 8, 21, 1, 8, 21, 2, 8, 21, 3, 8, 21, 1, 3, 8, 21, 34, 1, 34, 2, 34, 3, 34, 1, 3, 34, 5, 34, 1, 5, 34, 2, 5, 34
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n has A007895(n) terms.
With the 2nd Maple program, B(n) yields the number of terms in the Zeckendorf expansion of n, while Z(n) yields the expansion itself. For example, B(100)=3 and Z(100)=3, 8, 89. [Emeric Deutsch, Jul 05 2010]
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REFERENCES
| D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
Zeckendorf, E., Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972.
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LINKS
| T. D. Noe, Rows n=0..1000 of triangle, flattened
N. J. A. Sloane, Classic Sequences
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EXAMPLE
| 0=0; 1=1; 2=2; 3=3; 4=1+3; 5=5; 6=1+5; 7=2+5; 8=8; 9=1+8; 10=2+8; ... so triangle begins
0
1
2
3
1 3
5
1 5
2 5
8
1 8
2 8
3 8
1 3 8
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MAPLE
| Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 05 2010: (Start)
with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: F := proc (n) local i: for i while fibonacci(i) <= n do fibonacci(i) end do end proc: Z := proc (n) local j, z: for j to B(n) do z[j] := F(n-add(z[i], i = 1 .. j-1)) end do: seq(z[B(n)+1-k], k = 1 .. B(n)) end proc: for n to 25 do Z(n) end do;
# yields sequence in triangular form; end of this Maple program
with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: F := proc (n) local i; for i while fibonacci(i) <= n do fibonacci(i) end do end proc: Z := proc (n) local j, z: for j to B(n) do z[j] := F(n-add(z[i], i = 1 .. j-1)) end do: seq(z[B(n)+1-k], k = 1 .. B(n)) end proc: #
(End)
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MATHEMATICA
| f[n_] := (k=1; ff={}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); ro[n_] := If[n == 0, 0, r = n; s = {}; fr = f[n];
While[r > 0, lf = Last[fr]; If[lf <= r, r = r - lf; PrependTo[s, lf]]; fr = Drop[fr, -1]]; s]; Flatten[ro /@ Range[0, 42]] (* From Jean-François Alcover, Jul 23 2011 *)
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CROSSREFS
| Cf. A014417, A007895, A035514, A035515, A035516.
Sequence in context: A050375 A154722 A194760 * A099471 A121775 A127951
Adjacent sequences: A035514 A035515 A035516 * A035518 A035519 A035520
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KEYWORD
| nonn,easy,tabf,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 13 1999
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