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A194856 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(5).

%I A194856 #6 Mar 30 2012 18:57:44
%S A194856 1,2,1,3,2,1,4,3,2,1,4,3,2,1,5,4,3,2,6,1,5,4,3,7,2,6,1,5,4,8,3,7,2,6,
%T A194856 1,5,4,8,3,7,2,6,1,5,9,4,8,3,7,2,6,10,1,5,9,4,8,3,7,11,2,6,10,1,5,9,4,
%U A194856 8,12,3,7,11,2,6,10,1,5,9,4,8,12,3,7,11,2,6,10,1,5,9,13,4,8,12
%N A194856 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(5).
%C A194856 See A194832 for a general discussion.
%e A194856 First nine rows:
%e A194856 1
%e A194856 2 1
%e A194856 3 2 1
%e A194856 4 3 2 1
%e A194856 4 3 2 1 5
%e A194856 4 3 2 6 1 5
%e A194856 4 3 7 2 6 1 5
%e A194856 4 8  3 7 2 6 1 5
%e A194856 4 8  3 7 2 6 1 5  9
%t A194856 r = -Sqrt[5];
%t A194856 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194856 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t A194856 Sort[t[n], Less]], {n, 1, 20}]]  (* A194856 *)
%t A194856 TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t A194856 Sort[t[n], Less]], {n, 1, 15}]]
%t A194856 row[n_] := Position[f, n];
%t A194856 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194856 g[n_, k_] := Part[row[n], k];
%t A194856 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A194856  {k, 1, n}]]  (* A194857 *)
%t A194856 q[n_] := Position[p, n]; Flatten[Table[q[n],
%t A194856 {n, 1, 80}]]  (* A194858 *)
%Y A194856 Cf. A194832, A194857, A194858.
%K A194856 nonn,tabl
%O A194856 1,2
%A A194856 _Clark Kimberling_, Sep 04 2011