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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., 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.

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

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Dec 13 1999

STATUS

approved

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Last modified December 13 17:30 EST 2017. Contains 295959 sequences.