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 A035516 Triangular array formed from Zeckendorf expansion of integers: repeatedly subtract the largest Fibonacci number you can until nothing remains. 14
 0, 1, 2, 3, 3, 1, 5, 5, 1, 5, 2, 8, 8, 1, 8, 2, 8, 3, 8, 3, 1, 13, 13, 1, 13, 2, 13, 3, 13, 3, 1, 13, 5, 13, 5, 1, 13, 5, 2, 21, 21, 1, 21, 2, 21, 3, 21, 3, 1, 21, 5, 21, 5, 1, 21, 5, 2, 21, 8, 21, 8, 1, 21, 8, 2, 21, 8, 3, 21, 8, 3, 1, 34, 34, 1, 34, 2, 34, 3, 34, 3, 1, 34, 5, 34, 5, 1, 34, 5, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n has A007895(n) terms. 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 N. J. A. Sloane, Classic Sequences EXAMPLE 16 = 13 + 3, so row 16 is 13, 6. The first few rows are:   0;   1;   2;   3;   3, 1;   5;   5, 1;   5, 2;   8;   8, 1;   8, 2;   ... Row 1000000 is 832040,121393,46368,144,55. Indeed, the Maple program yields in no time Z(1000000) = {55,144,46368,121393,832040}. - Emeric Deutsch, Oct 22 2014 MAPLE with(combinat): Z := proc (n) local F, LF, A, m: F := proc (n) options operator, arrow: fibonacci(n) end proc: LF := proc (m) local i: for i from 0 while F(i) <= m do  end do: F(i-1) end proc: A := {}: m := n: while 0 < m do A := `union`(A, {LF(m)}): m := m-LF(m) end do: A end proc: # The Maple program, with the command Z(n), yields the set of the Fibonacci numbers in the Zeckendorf representation of n (terms in {} are in reverse order). - Emeric Deutsch, Oct 21 2014 MATHEMATICA t = Fibonacci /@ Range@ 12; Table[If[MemberQ[t, n], {n}, Most@ MapAt[# + 1 &, Abs@ Differences@ FixedPointList[# - First@ Reverse@ TakeWhile[t, Function[k, # >= k]] &, n], -1]], {n, 41}] // Flatten (* faster, or *) t = Fibonacci /@ Range@ 12; {{0}}~Join~Table[First@ Select[ Select[ IntegerPartitions@ n, Times @@ Boole@ Map[MemberQ[t, #] &, #] == 1 &], Times @@ Boole@ Map[# > 1 &, Abs@ Differences@ Map[Position[t, #][[1, 1]] &, #, {1}]] == 1 &], {n, 41}] // Flatten (* Michael De Vlieger, May 17 2016 *) PROG (Haskell) a035516 n k = a035516_tabf !! n !! k a035516_tabf = map a035516_row [0..] a035516_row 0 = [0] a035516_row n = z n \$ reverse \$ takeWhile (<= n) a000045_list where    z 0 _              = []    z x (f:fs'@(_:fs)) = if f <= x then f : z (x - f) fs else z x fs' -- Reinhard Zumkeller, Mar 10 2013 CROSSREFS Cf. A035517, A035514, A035515, A000045, A106530, A273156. Sequence in context: A077990 A085667 A220114 * A120428 A079950 A174953 Adjacent sequences:  A035513 A035514 A035515 * A035517 A035518 A035519 KEYWORD nonn,easy,tabf AUTHOR EXTENSIONS More terms from James A. Sellers, Dec 13 1999 STATUS approved

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Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)