A194868 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, 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, 8, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 9, 6, 3, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 8, 5, 2, 10, 7, 4, 1, 9, 6, 3, 11, 8, 5, 2, 10, 7, 4, 12, 1, 9, 6, 3, 11, 8, 5, 13, 2, 10, 7, 4, 12, 1, 9, 6, 3, 11, 8, 5, 13

Offset: 1

Author

Clark Kimberling, Sep 04 2011

Keywords

• nonn
• tabl

Comments

See A194832 for a general discussion.

Examples

			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
8 5 2 7 4 1 6 3
8 5 2 7 4 1 9 6 3
		

Crossrefs

Cf. A194832, A194869, A194870.

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}]]  (* A194868 *)
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}]] (* A194869 *)
q[n_] := Position[p, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A194870 *)