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 A035517 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. 33
 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; text; internal format)
 OFFSET 0,3 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] REFERENCES Zeckendorf, E., 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 T. D. Noe, Rows n=0..1000 of triangle, flattened D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60. N. J. A. Sloane, Classic Sequences 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 MAPLE 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; # Emeric Deutsch, Jul 05 2010 # yields sequence in triangular form; end of this Maple program 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]] (* Jean-François Alcover, Jul 23 2011 *) PROG (Haskell) a035517 n k = a035517_tabf !! n !! k a035517_row n = a035517_tabf !! n a035517_tabf = map reverse a035516_tabf -- Reinhard Zumkeller, Mar 10 2013 CROSSREFS Cf. A014417, A007895, A035514, A035515, A035516. Sequence in context: A300724 A154722 A194760 * A099471 A243574 A121775 Adjacent sequences:  A035514 A035515 A035516 * A035518 A035519 A035520 KEYWORD nonn,easy,tabf,nice,look AUTHOR EXTENSIONS More terms from James A. Sellers, Dec 13 1999 STATUS approved

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Last modified December 17 14:12 EST 2018. Contains 318201 sequences. (Running on oeis4.)