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%I
%S 1,1,2,3,1,2,3,1,4,2,5,3,1,4,2,5,3,1,6,4,2,7,5,3,1,6,4,2,7,5,3,1,8,6,
%T 4,2,9,7,5,3,1,8,6,4,2,9,7,5,3,1,10,8,6,4,2,11,9,7,5,3,1,10,8,6,4,2,
%U 11,9,7,5,3,1,12,10,8,6,4,2,13,11,9,7,5,3,1,12,10,8,6,4,2
%N 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(12).
%C See A194832 for a general discussion.
%e First nine rows:
%e 1
%e 1 2
%e 3 1 2
%e 3 1 4 2
%e 5 3 1 4 2
%e 5 3 1 6 4 2
%e 7 5 3 1 6 4 2
%e 7 5 3 1 8 6 4 2
%e 9 7 5 3 1 8 6 4 2
%t r = Sqrt[12];
%t t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t Sort[t[n], Less]], {n, 1, 20}]] (* A194899 *)
%t TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t Sort[t[n], Less]], {n, 1, 15}]]
%t row[n_] := Position[f, n];
%t u = TableForm[Table[row[n], {n, 1, 20}]]
%t g[n_, k_] := Part[row[n], k];
%t p = Flatten[Table[g[k, n - k + 1], {n, 1, 15},
%t {k, 1, n}]] (* A194900 *)
%t q[n_] := Position[p, n]; Flatten[Table[q[n],
%t {n, 1, 90}]] (* A194901 *)
%o (PARI) row(n) = Vec(vecsort(vector(n, k, frac(k*sqrt(12))),,1));
%o tabl(nn) = for (n=1, nn, print(row(n))); \\ _Michel Marcus_, Feb 06 2019
%Y Cf. A194832, A194900, A194901.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Sep 05 2011
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