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A003603
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Fractal sequence obtained from Fibonacci numbers (or Wythoff array).
(Formerly M0138)
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52
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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]
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REFERENCES
| C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Reinhard Zumkeller, Rows n=1..20 of triangle, flattened
C. Kimberling, Fractal sequences
N. J. A. Sloane, Classic Sequences
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FORMULA
| Vertical para-budding sequence: says which row of Wythoff array (starting row count at 1) contains n.
If delete first occurrence of 1, 2, 3, ... the sequence is unchanged.
Contribution from Clark Kimberling (ck6(AT)evansville.edu), 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 concatentation of the rows. (End)
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EXAMPLE
| Contribution from Clark Kimberling (ck6(AT)evansville.edu), Oct 29 2009: (Start)
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. (End)
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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}] (* Gyorgy Birkas, Apr 13, 2011 *)
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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
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CROSSREFS
| Equals A019586(n) + 1. Cf. A003602.
Sequence in context: A167287 A007336 A133334 * A188277 A135227 A104325
Adjacent sequences: A003600 A003601 A003602 * A003604 A003605 A003606
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KEYWORD
| nonn,easy,nice,eigen,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
Keyword tabf added by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2012
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