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A194905 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 A194905 #9 Apr 11 2015 10:08:06
%S A194905 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,1,2,3,4,5,6,7,8,1,2,3,4,5,
%T A194905 6,7,8,1,9,2,3,4,5,6,7,8,1,9,2,10,3,4,5,6,7,8,1,9,2,10,3,11,4,5,6,7,8,
%U A194905 1,9,2,10,3,11,4,12,5,6,7,8,1,9,2,10,3,11,4,12,5,13,6,7,8,1,9
%N A194905 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 A194905 See A194832 for a general discussion.
%e A194905 First nine rows:
%e A194905 1
%e A194905 1 2
%e A194905 1 2 3
%e A194905 1 2 3 4
%e A194905 1 2 3 4 5
%e A194905 1 2 3 4 5 6
%e A194905 1 2 3 4 5 6 7
%e A194905 8 1 2 3 4 5 6 7
%e A194905 8 1 9 2 3 4 5 6 7
%t A194905 r = Pi;
%t A194905 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194905 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t A194905 Sort[t[n], Less]], {n, 1, 20}]]  (* A194905 *)
%t A194905 TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t A194905 Sort[t[n], Less]], {n, 1, 15}]]
%t A194905 row[n_] := Position[f, n];
%t A194905 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194905 g[n_, k_] := Part[row[n], k];
%t A194905 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A194905 {k, 1, n}]]  (* A194906 *)
%t A194905 q[n_] := Position[p, n]; Flatten[Table[q[n],
%t A194905 {n, 1, 80}]]  (* A194907 *)
%Y A194905 Cf. A194832, A194906, A194907.
%K A194905 nonn,tabl
%O A194905 1,3
%A A194905 _Clark Kimberling_, Sep 05 2011