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A194899 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). 4

%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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Last modified February 5 08:15 EST 2023. Contains 360082 sequences. (Running on oeis4.)