A191454 Dispersion of (2*floor(n*r)), where r=(golden ratio), by antidiagonals.

1, 2, 3, 6, 8, 4, 18, 24, 12, 5, 58, 76, 38, 16, 7, 186, 244, 122, 50, 22, 9, 600, 788, 394, 160, 70, 28, 10, 1940, 2550, 1274, 516, 226, 90, 32, 11, 6276, 8250, 4122, 1668, 730, 290, 102, 34, 13, 20308, 26696, 13338, 5396, 2362, 938, 330, 110, 42, 14, 65718

Offset: 1

Author

Clark Kimberling, Jun 05 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....6....18...58
  3...8....24...76...244
  4...12...38...122..394
  5...16...50...160..516
  7...22...70...226..730
		

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_] :=2Floor[n*x]   (* 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}]]
(* A191454 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191454 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)