A054065 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),...

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, 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, 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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			p(1)=(1); p(2)=(2,1); p(3)=(2,1,3); p(4)=(2,4,1,3).
As a triangular array (see A194832), first nine rows:
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 3 8
5 2 7 4 9 1 6 3 8
		

Crossrefs

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.

Programs

Mathematica

r = (1 + Sqrt[5])/2;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]] (* A054065 *)
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}]] (* A054069 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A054068 *)
(* Clark Kimberling, Sep 03 2011 *)
Flatten[Table[Ordering[Table[FractionalPart[GoldenRatio k], {k, n}]], {n, 10}]] (* Birkas Gyorgy, Jun 30 2012 *)

Extensions

Extended by Ray Chandler, Apr 18 2009