login
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=-sqrt(6).
4

%I #5 Mar 30 2012 18:57:44

%S 1,2,1,2,1,3,2,4,1,3,2,4,1,3,5,2,4,6,1,3,5,2,4,6,1,3,5,7,2,4,6,8,1,3,

%T 5,7,2,4,6,8,1,3,5,7,9,2,4,6,8,10,1,3,5,7,9,11,2,4,6,8,10,1,3,5,7,9,

%U 11,2,4,6,8,10,1,12,3,5,7,9,11,2,13,4,6,8,10,1,12,3,5,7,9,11,2,13

%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=-sqrt(6).

%C See A194832 for a general discussion.

%e First nine rows:

%e 1

%e 2 1

%e 2 1 3

%e 2 4 1 3

%e 2 4 1 3 5

%e 2 4 6 1 3 5

%e 2 4 6 1 3 5 7

%e 2 4 6 8 1 3 5 7

%e 2 4 6 8 1 3 5 7 9

%t r = -Sqrt[6];

%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}]] (* A194874 *)

%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}]] (* A194875 *)

%t q[n_] := Position[p, n]; Flatten[

%t Table[q[n], {n, 1, 80}]] (* A194876 *)

%Y Cf. A194832, A194875, A194876.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Sep 04 2011