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

1,2

Links

Table of n, a(n) for n=1..99.

Example

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

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 *)

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.

Sequence in context: A194902 A194874 A194835 * A194868 A304574 A139024

Adjacent sequences: A054062 A054063 A054064 * A054066 A054067 A054068

Keyword

nonn

Author

Clark Kimberling

Extensions

Extended by Ray Chandler, Apr 18 2009

Status

approved