OFFSET

1,2

COMMENTS

Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

EXAMPLE

Northwest corner:

1, 2, 5, 14, 39, ...

3, 8, 22, 62, 175, ...

4, 11, 31, 87, 246, ...

6, 16, 45, 127, 359, ...

7, 19, 53, 149, 421, ...

MATHEMATICA

(* Program generates the dispersion array T of the increasing sequence f[n] *)

r=40; r1=12; c=40; c1=12; f[n_] :=Floor[2n*Sqrt[2]] (* complement of column 1 *)

mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

rows = {NestList[f, 1, c]};

Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

t[i_, j_] := rows[[i, j]];

TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A191540 array *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191540 sequence *)

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 06 2011

STATUS

approved