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 A189920 Zeckendorf representation of natural numbers. 10
 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS The row lengths sequence of this array is A072649(n), n >= 1. Note that the Fibonacci numbers F(0)=0 and F(1)=1 are not used in this unique representation of n >= 1. No neighboring Fibonacci numbers are allowed (no 1,1, subsequence in any row n). T(n,k) = A213676(n, A072649(n, k)-1) for k = 1..A072649(k). - 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 41.3-4 (1972) 179-182 (with the proof from 1939). R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., 1994, Addison-Wesley, Reading MA, pp. 295-296. LINKS Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened FORMULA n = Sum_{m=1..rl(n)} a(n,m)*F(rl(n) + 2 - m), n >= 1, with rl(n):=A072649(n)(row length) and F(n):=A000045(n) (Fibonacci numbers). EXAMPLE n=1:  1; n=2:  1, 0; n=3:  1, 0, 0; n=4:  1, 0, 1; n=5:  1, 0, 0, 0; n=6:  1, 0, 0, 1; n=7:  1, 0, 1, 0; n=8:  1, 0, 0, 0, 0; n=9:  1, 0, 0, 0, 1; n=10: 1, 0, 0, 1, 0; n=11: 1, 0, 1, 0, 0; n=12: 1, 0, 1, 0, 1; n=13: 1, 0, 0, 0, 0, 0; ... 1 = F(2), 6 = F(5) + F(2), 11 = F(6) + F(4). MATHEMATICA f[n_] := (k = 1; ff = {}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); a[n_] := (fn = f[n]; xx = Array[x, Length[fn]]; r = xx /. {ToRules[ Reduce[ And @@ (0 <= # <= 1 & ) /@ xx && fn . xx == n, xx, Integers]]}; Reverse[ First[ Select[ r, FreeQ[ Split[#], {1, 1, ___}] & ]]]); Flatten[ Table[ a[n], {n, 1, 25}]] (* Jean-François Alcover, Sep 29 2011 *) PROG (Haskell) a189920 n k = a189920_row n !! k a189920_row n = z n \$ reverse \$ takeWhile (<= n) \$ tail a000045_list where    z x (f:fs'@(_:fs)) | f == 1 = if x == 1 then [1] else []                       | f == x = 1 : replicate (length fs) 0                       | f < x  = 1 : 0 : z (x - f) fs                       | f > x  = 0 : z x fs' a189920_tabf = map a189920_row [1..] -- Reinhard Zumkeller, Mar 10 2013 CROSSREFS Cf. A035517, A014417. Sequence in context: A257234 A266666 A190191 * A295896 A275661 A266716 Adjacent sequences:  A189917 A189918 A189919 * A189921 A189922 A189923 KEYWORD nonn,easy,tabf,base AUTHOR Wolfdieter Lang, Jun 12 2011 STATUS approved

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Last modified November 18 20:13 EST 2018. Contains 317324 sequences. (Running on oeis4.)