A003603 Fractal sequence obtained from Fibonacci numbers (or Wythoff array). (Formerly M0138)

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

1,4

Comments

Length of n-th row = A000045(n); last term of n-th row = A094967(n-1); sum of n-th row = A033192(n-1). - Reinhard Zumkeller, Jan 26 2012

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.

N. J. A. Sloane, Classic Sequences.

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

Example

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

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

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 *)

Prog

(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

Crossrefs

Equals A019586(n) + 1. Cf. A003602, A000045, A033192, A035513, A035612, A094967, A265650.

Sequence in context: A007336 A227539 A133334 * A278118 A255238 A212536

Adjacent sequences: A003600 A003601 A003602 * A003604 A003605 A003606

Keyword

nonn,easy,nice,eigen,tabf

Author

N. J. A. Sloane, Mira Bernstein

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

Keyword tabf added by Reinhard Zumkeller, Jan 26 2012

Status

approved