A191540 Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.

1, 2, 3, 5, 8, 4, 14, 22, 11, 6, 39, 62, 31, 16, 7, 110, 175, 87, 45, 19, 9, 311, 494, 246, 127, 53, 25, 10, 879, 1397, 695, 359, 149, 70, 28, 12, 2486, 3951, 1965, 1015, 421, 197, 79, 33, 13, 7031, 11175, 5557, 2870, 1190, 557, 223, 93, 36, 15, 19886, 31607

Offset: 1

Author

Clark Kimberling, Jun 06 2011

Keywords

• nonn
• tabl

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:

(1) s=A000040 (the primes), D=A114537, u=A114538.

(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

More recent examples of dispersions: A191426-A191455 and A191536-A191545.

Examples

			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, ...
		

Links

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

Crossrefs

Cf. A114537, A035513, A035506.

Programs

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 *)