A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

1, 2, 3, 4, 7, 5, 9, 16, 12, 6, 21, 38, 28, 14, 8, 50, 91, 67, 33, 19, 10, 120, 219, 161, 79, 45, 24, 11, 289, 528, 388, 190, 108, 57, 26, 13, 697, 1274, 936, 458, 260, 137, 62, 31, 15, 1682, 3075, 2259, 1105, 627, 330, 149, 74, 36, 17, 4060, 7423, 5453

Offset: 1

Author

Clark Kimberling, Jun 04 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.

Examples

			Northwest corner:
  1....2....4....9....21
  3....7....16...38...91
  5....12...28...67...161
  6....14...33...79...190
  8....19...45...108..260
		

Crossrefs

Cf. A114537, A035513, A035506, A191438.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n] (* 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, 10}, {j, 1, 10}]]
(* A191438 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191438 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)