A191436 Dispersion of ([n*x+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

1, 4, 2, 12, 6, 3, 33, 17, 9, 5, 88, 46, 25, 14, 7, 232, 122, 67, 38, 19, 8, 609, 321, 177, 101, 51, 22, 10, 1596, 842, 465, 266, 135, 59, 27, 11, 4180, 2206, 1219, 698, 355, 156, 72, 30, 13, 10945, 5777, 3193, 1829, 931, 410, 190, 80, 35, 15, 28656, 15126

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....4....12...33...88
  2....6....17...46...122
  3....9....25...67...177
  5....14...38...101..266
  7....19...51...135..355
		

Crossrefs

Cf. A114537, A035513, A035506.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;  x = GoldenRatio;
f[n_] := Floor[n*x+n+x-1] (* 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}]]
(* A191436 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191436 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)