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(8).
4

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

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

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

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

%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(8).

%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 5 4 3 2 1 6

%e 5 4 3 2 7 1 6

%e 5 4 3 8 2 7 1 6

%e 5 4 9 3 8 2 7 1 6

%t r = Sqrt[8];

%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}]] (* A194877 *)

%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}]] (* A194878 *)

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

%t {n, 1, 80}]] (* A194879 *)

%Y Cf. A194832, A194878, A194879.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Sep 04 2011