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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.

%I A194832 #33 Jul 30 2023 02:41:21
%S A194832 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,
%T A194832 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,
%U A194832 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
%N 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.
%C A194832 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.
%C A194832 Examples:
%C A194832 r................Array1....Array2....Perm2
%C A194832 tau..............A054065...A054069...A054068
%C A194832 -tau.............A194832...A194833...A194834
%C A194832 sqrt(2)..........A054073...A054077...A054076
%C A194832 -sqrt(2).........A194835...A194836...A194837
%C A194832 sqrt(3)..........A194838...A194839...A194840
%C A194832 -sqrt(3).........A194841...A194842...A194843
%C A194832 sqrt(5)..........A194844...A194845...A194846
%C A194832 -sqrt(5).........A194856...A194857...A194858
%C A194832 sqrt(6)..........A194871...A194872...A194873
%C A194832 -sqrt(6).........A194874...A194875...A194876
%C A194832 sqrt(8)..........A194877...A194878...A194879
%C A194832 -sqrt(8).........A194896...A194897...A194898
%C A194832 sqrt(12).........A194899...A194900...A194901
%C A194832 -sqrt(12)........A194902...A194903...A194904
%C A194832 e................A194859...A194860...A194861
%C A194832 -e...............A194865...A194866...A194864
%C A194832 pi...............A194905...A194906...A194907
%C A194832 -pi..............A194908...A194909...A194910
%C A194832 (1+sqrt(3))/2....A194862...A194863...A194867
%C A194832 -(1+sqrt(3))/2...A194868...A194869...A194870
%C A194832 2^(1/3)..........A194911...A194912...A194913
%D A194832 C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997), 157-168.
%H A194832 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fractal_sequence">Fractal sequence</a>
%e A194832 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]
%e A194832 First nine rows:
%e A194832   1
%e A194832   1 2
%e A194832   3 1 2
%e A194832   3 1 4 2
%e A194832   3 1 4 2 5
%e A194832   3 6 1 4 2 5
%e A194832   3 6 1 4 7 2 5
%e A194832   8 3 6 1 4 7 2 5
%e A194832   8 3 6 1 9 4 7 2 5
%t A194832 r = -GoldenRatio;
%t A194832 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194832 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]]
%t A194832 (* A194832 *)
%t A194832 TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
%t A194832 row[n_] := Position[f, n];
%t A194832 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194832 g[n_, k_] := Part[row[n], k];
%t A194832 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194833 *)
%t A194832 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194834 *)
%Y A194832 Cf. A194833, A194834, A054065.
%K A194832 nonn,tabl
%O A194832 1,3
%A A194832 _Clark Kimberling_, Sep 03 2011
%E A194832 Table in overview corrected by _Georg Fischer_, Jul 30 2023