A194874 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=-sqrt(6).

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

Crossrefs

Cf. A194832, A194875, A194876.

Programs

Mathematica

r = -Sqrt[6];
t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
Sort[t[n], Less]], {n, 1, 20}]]   (* A194874 *)
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}]]   (* A194875 *)
q[n_] := Position[p, n]; Flatten[
Table[q[n], {n, 1, 80}]]   (* A194876 *)