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A194865 Triangular array (and fractal sequence): row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {-e}, {-2e}, ..., {-ne}.

%I A194865 #8 Jul 30 2023 03:35:45
%S A194865 1,1,2,1,2,3,4,1,2,3,4,1,5,2,3,4,1,5,2,6,3,4,1,5,2,6,3,7,4,8,1,5,2,6,
%T A194865 3,7,4,8,1,5,9,2,6,3,7,4,8,1,5,9,2,6,10,3,7,11,4,8,1,5,9,2,6,10,3,7,
%U A194865 11,4,8,1,12,5,9,2,6,10,3,7,11,4,8,1,12,5,9,2,13,6,10,3,7,11,4,8
%N A194865 Triangular array (and fractal sequence):  row n is the permutation of (1,2,...,n) obtained from the increasing ordering of fractional parts {-e}, {-2e}, ..., {-ne}.
%C A194865 See A194832 for a general discussion.
%e A194865 First nine rows:
%e A194865 1
%e A194865 1 2
%e A194865 1 2 3
%e A194865 4 1 2 3
%e A194865 4 1 5 2 3
%e A194865 4 1 5 2 6 3
%e A194865 4 1 5 2 6 3 7
%e A194865 4 8 1 5 2 6 3 7
%e A194865 4 8 1 5 9 2 6 3 7
%t A194865 r = -E;
%t A194865 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A194865 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@
%t A194865 Sort[t[n], Less]], {n, 1, 20}]]  (* A194865 *)
%t A194865 TableForm[Table[Flatten[(Position[t[n], #1] &) /@
%t A194865 Sort[t[n], Less]], {n, 1, 15}]]
%t A194865 row[n_] := Position[f, n];
%t A194865 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A194865 g[n_, k_] := Part[row[n], k];
%t A194865 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A194865 {k, 1, n}]]  (* A194866 *)
%t A194865 q[n_] := Position[p, n]; Flatten[Table[q[n],
%t A194865 {n, 1, 80}]]  (* A194864 *)
%Y A194865 Cf. A194832, A194864, A194866.
%K A194865 nonn,tabl
%O A194865 1,3
%A A194865 _Clark Kimberling_, Sep 04 2011