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 A194832 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. 45
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 LINKS Wikipedia, Fractal sequence EXAMPLE Fractional parts: {r}=0.3...; {2r}=0.7...; {3r}=0.1...; {4r}=0.5...; thus, row 4 is (3,1,4,2) because {3r} < {r} < {4r} < {2r}. 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 MATHEMATICA 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 *) CROSSREFS Cf. A194833, A194834, A054065. Sequence in context: A175469 A239526 A194862 * A195107 A054073 A194871 Adjacent sequences:  A194829 A194830 A194831 * A194833 A194834 A194835 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Sep 03 2011 STATUS approved

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Last modified December 12 03:27 EST 2018. Contains 318052 sequences. (Running on oeis4.)