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A194908 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=-Pi.

%I A194908 #8 Apr 10 2021 02:05:07
%S A194908 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,7,6,5,4,3,2,
%T A194908 1,8,7,6,5,4,3,2,9,1,8,7,6,5,4,3,10,2,9,1,8,7,6,5,4,11,3,10,2,9,1,8,7,
%U A194908 6,5,12,4,11,3,10,2,9,1,8,7,6,13,5,12,4,11,3,10,2,9,1,8,7,14,6
%N A194908 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=-Pi.
%C A194908 See A194832 for a general discussion.
%e A194908 First nine rows:
%e A194908   1;
%e A194908   2, 1;
%e A194908   3, 2, 1;
%e A194908   4, 3, 2, 1;
%e A194908   5, 4, 3, 2, 1;
%e A194908   6, 5, 4, 3, 2, 1;
%e A194908   7, 6, 5, 4, 3, 2, 1;
%e A194908   7, 6, 5, 4, 3, 2, 1, 8;
%e A194908   7, 6, 5, 4, 3, 2, 9, 1, 8;
%t A194908 r = -Pi;
%t A194908 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194908 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t A194908 Sort[t[n], Less]], {n, 1, 20}]]  (* A194908 *)
%t A194908 TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t A194908 Sort[t[n], Less]], {n, 1, 15}]]
%t A194908 row[n_] := Position[f, n];
%t A194908 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194908 g[n_, k_] := Part[row[n], k];
%t A194908 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A194908 {k, 1, n}]]  (* A194909 *)
%t A194908 q[n_] := Position[p, n]; Flatten[Table[q[n],
%t A194908 {n, 1, 80}]]  (* A194910 *)
%Y A194908 Cf. A194832, A194909, A194910.
%K A194908 nonn,tabl
%O A194908 1,2
%A A194908 _Clark Kimberling_, Sep 05 2011