A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

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

Offset

1,3

Comments

A054073 generates the interspersion A054077; see A194832 and the Mathematica program.

Links

Table of n, a(n) for n=1..86.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Example

p(1)=(1); p(2)=(1,2); p(3)=(3,1,2); p(4)=(3,1,4,2).
When formatted as a triangle, the first 9 rows:
1
1 2
3 1 2
3 1 4 2
5 3 1 4 2
5 3 1 6 4 2
5 3 1 6 4 2 7
5 3 8 1 6 4 2 7
5 3 8 1 6 4 9 2 7

Mathematica

r = Sqrt[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}]] (* A054073 *)
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}]] (* A054077 *)
q[n_] := Position[p, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A054076 *)
(* Clark Kimberling, Sep 03 2011 *)

Crossrefs

Cf. A054071, A054072, A194832.

Sequence in context: A194862 A194832 A195107 * A194871 A194899 A228094

Adjacent sequences: A054070 A054071 A054072 * A054074 A054075 A054076

Keyword

nonn

Author

Clark Kimberling

Status

approved