A003603 Fractal sequence obtained from Fibonacci numbers (or Wythoff array).
1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 1, 6, 4, 3, 7, 2, 8, 5, 1, 9, 6, 4, 10, 3, 11, 7, 2, 12, 8, 5, 13, 1, 14, 9, 6, 15, 4, 16, 10, 3, 17, 11, 7, 18, 2, 19, 12, 8, 20, 5, 21, 13, 1, 22, 14, 9, 23, 6, 24, 15, 4, 25, 16, 10, 26, 3, 27, 17, 11, 28, 7, 29, 18, 2, 30, 19, 12, 31, 8, 32, 20, 5, 33
Offset: 1
Examples
In the recurrence for making new rows, we get row 5 from row 4 thus: write row 4: 1,3,2, and then place 4 right after 1, and place 5 right after 2, getting 1,4,3,2,5. - _Clark Kimberling_, Oct 29 2009
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Reinhard Zumkeller, Rows n = 1..20 of triangle, flattened
- J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
- David Garth and Joseph Palmer, Self-Similar Sequences and Generalized Wythoff Arrays, Fibonacci Quart. 54 (2016), no. 1, 72-78.
- Clark Kimberling, Fractal sequences.
- Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
- A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 8.
- N. J. A. Sloane, Classic Sequences.
Programs
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Haskell
-- according to Kimberling, see formula section. a003603 n k = a003603_row n !! (k-1) a003603_row n = a003603_tabl !! (n-1) a003603_tabl = [1] : [1] : wythoff [2..] [1] [1] where wythoff is xs ys = f is xs ys [] where f js [] [] ws = ws : wythoff js ys ws f js [] [v] ws = f js [] [] (ws ++ [v]) f (j:js) (u:us) (v:vs) ws | u == v = f js us vs (ws ++ [v,j]) | u /= v = f (j:js) (u:us) vs (ws ++ [v]) -- Reinhard Zumkeller, Jan 26 2012 -
Maple
A003603 := proc(n::posint) local r,c,W ; for r from 1 do for c from 1 do W := A035513(r,c) ; if W = n then return r ; elif W > n then break ; end if; end do: end do: end proc: seq(A003603(n),n=1..100) ; # R. J. Mathar, Aug 13 2021
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Mathematica
num[n_, b_] := Last[NestWhile[{Mod[#[[1]], Last[#[[2]]]], Drop[#[[2]], -1], Append[#[[3]], Quotient[#[[1]], Last[#[[2]]]]]} &, {n, b, {}}, #[[2]] =!= {} &]]; left[n_, b_] := If[Last[num[n, b]] == 0, Dot[num[n, b], Rest[Append[Reverse[b], 0]]], n]; fractal[n_, b_] := # - Count[Last[num[Range[#], b]], 0] &@ FixedPoint[left[#, b] &, n]; Table[fractal[n, Table[Fibonacci[i], {i, 2, 12}]], {n, 30}] (* Birkas Gyorgy, Apr 13 2011 *) row[1] = row[2] = {1}; row[n_] := row[n] = Module[{ro, pos, lp, ins}, ro = row[n-1]; pos = Position[ro, Alternatives @@ Intersection[ro, row[n-2]]] // Flatten; lp = Length[pos]; ins = Range[lp] + Max[ro]; Do[ro = Insert[ro, ins[[i]], pos[[i]] + i], {i, 1, lp}]; ro]; Array[row, 9] // Flatten (* Jean-François Alcover, Jul 12 2016 *)
Formula
Vertical para-budding sequence: says which row of Wythoff array (starting row count at 1) contains n.
If one deletes the first occurrence of 1, 2, 3, ... the sequence is unchanged.
From Clark Kimberling, Oct 29 2009: (Start)
The fractal sequence of the Wythoff array can be constructed without reference to the Wythoff array or Fibonacci numbers. Write initial rows:
Row 1: .... 1
Row 2: .... 1
Row 3: .... 1..2
Row 4: .... 1..3..2
For n>4, to form row n+1, let k be the least positive integer not yet used; write row n, and right after the first number that is also in row n-1, place k; right after the next number that is also in row n-1, place k+1, and continue. A003603 is the concatenation of the rows. (End)
Conjecture: a(n) = abs(floor(n/phi) - floor(n*(1/phi + 1/(-phi)^(A035612(n) + 1)))) where phi = (1+sqrt(5))/2. - Alan Michael Gómez Calderón, Oct 27 2023
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
Keyword tabf added by Reinhard Zumkeller, Jan 26 2012
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