A054065 - OEIS

%I #26 Apr 15 2014 14:15:51

%S 1,2,1,2,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,7,4,1,6,3,5,2,7,4,1,6,

%T 3,8,5,2,7,4,9,1,6,3,8,5,10,2,7,4,9,1,6,3,8,5,10,2,7,4,9,1,6,11,3,8,5,

%U 10,2,7,12,4,9,1,6,11,3,8,13,5,10,2,7,12,4,9,1,6,11,3,8,13,5,10,2,7,12,4,9

%N Fractal sequence induced by tau: for k >= 1, let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*tau} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

%e p(1)=(1); p(2)=(2,1); p(3)=(2,1,3); p(4)=(2,4,1,3).

%e As a triangular array (see A194832), first nine rows:

%e 1

%e 2 1

%e 2 1 3

%e 2 4 1 3

%e 5 2 4 1 3

%e 5 2 4 1 6 3

%e 5 2 7 4 1 6 3

%e 5 2 7 4 1 6 3 8

%e 5 2 7 4 9 1 6 3 8

%t r = (1 + Sqrt[5])/2;

%t t[n_] := Table[FractionalPart[k*r], {k, 1, n}];

%t f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]] (* A054065 *)

%t TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]

%t row[n_] := Position[f, n];

%t u = TableForm[Table[row[n], {n, 1, 20}]]

%t g[n_, k_] := Part[row[n], k];

%t p = Flatten[Table[g[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A054069 *)

%t q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A054068 *)

%t (* _Clark Kimberling_, Sep 03 2011 *)

%t Flatten[Table[Ordering[Table[FractionalPart[GoldenRatio k], {k, n}]], {n, 10}]] (* _Birkas Gyorgy_, Jun 30 2012 *)

%Y Position of 1 in p(k) is given by A019446. Position of k in p(k) is given by A019587.  For related arrays and sequences, see A194832.

%K nonn

%O 1,2

%A _Clark Kimberling_

%E Extended by _Ray Chandler_, Apr 18 2009