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A194832
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Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {r}, {2r}, ..., {nr}, where r= -tau = -(1+sqrt(5))/2.
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45
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1, 1, 2, 3, 1, 2, 3, 1, 4, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3, 11
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OFFSET
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1,3
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COMMENTS
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Every irrational number r generates a triangular array in the manner exemplified here. Taken as a sequence, the numbers comprise a fractal sequence f which induces a second (rectangular) array whose n-th row gives the positions of n in f. Denote these by Array1 and Array2. As proved elsewhere, Array2 is an interspersion. (Every row intersperses every other row except for initial terms.) Taken as a sequence, Array2 is a permutation, Perm1, of the positive integers; let Perm2 denote its inverse permutation.
Examples:
r................Array1....Array2....Perm2
tau..............A054065...A054069...A054068
-tau.............A194832...A194833...A194834
sqrt(2)..........A054065...A054077...A054076
-sqrt(2).........A194835...A194836...A194837
sqrt(3)..........A194838...A194839...A194840
-sqrt(3).........A194841...A194842...A194843
sqrt(5)..........A194844...A194845...A194846
-sqrt(5).........A194856...A194857...A194858
sqrt(6)..........A194871...A194872...A194873
-sqrt(6).........A194874...A194875...A194876
sqrt(8)..........A194877...A194878...A194879
-sqrt(8).........A194896...A194897...A194898
sqrt(12).........A194899...A194900...A194901
-sqrt(12)........A194902...A194903...A194904
e................A194859...A194860...A194861
-e...............A194865...A194866...A194867
pi...............A194905...A194906...A194907
-pi..............A194908...A194909...A194910
(1+sqrt(3)/)/2...A194862...A194863...A194864
-(1+sqrt(3)/)/2..A194868...A194869...A194870
2^(1/3)..........A194911...A194912...A194913
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REFERENCES
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C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997), 157-168.
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LINKS
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Table of n, a(n) for n=1..94.
Wikipedia, Fractal sequence
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EXAMPLE
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Fractional parts: {-r}=-0.61..;{-2r}=-0.23..;{-3r}=-0.85..;{-4r}=-0.47..; thus, row 4 is (3,1,4,2) because {-3r} < {-r} < {-4r} < {-2r}. [corrected by Michel Dekking, Nov 30 2020]
First nine rows:
1
1 2
3 1 2
3 1 4 2
3 1 4 2 5
3 6 1 4 2 5
3 6 1 4 7 2 5
8 3 6 1 4 7 2 5
8 3 6 1 9 4 7 2 5
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MATHEMATICA
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r = -GoldenRatio;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]
(* A194832 *)
TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
row[n_] := Position[f, n];
u = TableForm[Table[row[n], {n, 1, 20}]]
g[n_, k_] := Part[row[n], k];
p = Flatten[Table[g[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194833 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194834 *)
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CROSSREFS
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Cf. A194833, A194834, A054065.
Sequence in context: A175469 A239526 A194862 * A195107 A054073 A194871
Adjacent sequences: A194829 A194830 A194831 * A194833 A194834 A194835
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Sep 03 2011
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STATUS
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approved
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