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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.

%I A194862 #7 Jul 30 2023 03:44:02
%S A194862 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,
%T A194862 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,
%U A194862 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
%N 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.
%C A194862 See A194832 for a general discussion.
%e A194862 First nine rows:
%e A194862 1
%e A194862 1 2
%e A194862 3 1 2
%e A194862 3 1 4 2
%e A194862 3 1 4 2 5
%e A194862 3 6 1 4 2 5
%e A194862 3 6 1 4 7 2 5
%e A194862 3 6 1 4 7 2 5 8
%e A194862 3 6 9 1 4 7 2 5 8
%t A194862 r = (1 + Sqrt[3])/2;
%t A194862 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194862 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t A194862 Sort[t[n], Less]], {n, 1, 20}]]  (* A194862 *)
%t A194862 TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t A194862 Sort[t[n], Less]], {n, 1, 15}]]
%t A194862 row[n_] := Position[f, n];
%t A194862 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194862 g[n_, k_] := Part[row[n], k];
%t A194862 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A194862 {k, 1, n}]]  (* A194863 *)
%t A194862 q[n_] := Position[p, n]; Flatten[Table[q[n],
%t A194862 {n, 1, 80}]]  (* A194867 *)
%Y A194862 Cf. A194832, A194863, A194867.
%K A194862 nonn,tabl
%O A194862 1,3
%A A194862 _Clark Kimberling_, Sep 04 2011