%I #8 Apr 10 2021 02:05:07
%S 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 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 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 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 See A194832 for a general discussion.
%e First nine rows:
%e 1;
%e 2, 1;
%e 3, 2, 1;
%e 4, 3, 2, 1;
%e 5, 4, 3, 2, 1;
%e 6, 5, 4, 3, 2, 1;
%e 7, 6, 5, 4, 3, 2, 1;
%e 7, 6, 5, 4, 3, 2, 1, 8;
%e 7, 6, 5, 4, 3, 2, 9, 1, 8;
%t r = -Pi;
%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}]] (* A194908 *)
%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}]] (* A194909 *)
%t q[n_] := Position[p, n]; Flatten[Table[q[n],
%t {n, 1, 80}]] (* A194910 *)
%Y Cf. A194832, A194909, A194910.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Sep 05 2011