A194862 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=(1+sqrt(3))/2.

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

Offset: 1

Author

Clark Kimberling, Sep 04 2011

Keywords

• nonn
• tabl

Comments

See A194832 for a general discussion.

Examples

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

Crossrefs

Cf. A194832, A194863, A194867.

Programs

Mathematica

r = (1 + Sqrt[3])/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}]]  (* A194862 *)
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}]]  (* A194863 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194867 *)