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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(5).
4

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

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

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

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

%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(5).

%C See A194832 for a general discussion.

%e First nine rows:

%e 1

%e 1 2

%e 1 2 3

%e 1 2 3 4

%e 5 1 2 3 4

%e 5 1 6 2 3 4

%e 5 1 6 2 7 3 4

%e 5 1 6 2 7 3 8 4

%e 9 5 1 6 2 7 3 8 4

%t r = Sqrt[5];

%t t[n_] := Table[FractionalPart[k*r], {k, 1, n}];

%t f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 20}]] (* A194844 *)

%t TableForm[Table[Flatten[(Position[t[n], #1] &) /@ 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}]] (* A194845 *)

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

%t Table[q[n], {n, 1, 80}]] (* A194846 *)

%Y Cf. A194832, A194845, A194846.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Sep 04 2011