A194908 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=-Pi.

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 9, 1, 8, 7, 6, 5, 4, 3, 10, 2, 9, 1, 8, 7, 6, 5, 4, 11, 3, 10, 2, 9, 1, 8, 7, 6, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8, 7, 6, 13, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8, 7, 14, 6

Offset: 1

Author

Clark Kimberling, Sep 05 2011

Keywords

• nonn
• tabl

Comments

See A194832 for a general discussion.

Examples

			First nine rows:
  1;
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  6, 5, 4, 3, 2, 1;
  7, 6, 5, 4, 3, 2, 1;
  7, 6, 5, 4, 3, 2, 1, 8;
  7, 6, 5, 4, 3, 2, 9, 1, 8;
		

Crossrefs

Cf. A194832, A194909, A194910.

Programs

Mathematica

r = -Pi;
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]  (* A194908 *)
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}]]  (* A194909 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194910 *)