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%I #7 Jul 30 2023 03:44:02
%S 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 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 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 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 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 3 1 4 2 5
%e 3 6 1 4 2 5
%e 3 6 1 4 7 2 5
%e 3 6 1 4 7 2 5 8
%e 3 6 9 1 4 7 2 5 8
%t r = (1 + Sqrt[3])/2;
%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}]] (* A194862 *)
%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, 13},
%t {k, 1, n}]] (* A194863 *)
%t q[n_] := Position[p, n]; Flatten[Table[q[n],
%t {n, 1, 80}]] (* A194867 *)
%Y Cf. A194832, A194863, A194867.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Sep 04 2011
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