A191542 Dispersion of (2*floor(n*sqrt(3))), by antidiagonals.

1, 2, 3, 6, 10, 4, 20, 34, 12, 5, 68, 116, 40, 16, 7, 234, 400, 138, 54, 24, 8, 810, 1384, 478, 186, 82, 26, 9, 2804, 4794, 1654, 644, 284, 90, 30, 11, 9712, 16606, 5728, 2230, 982, 310, 102, 38, 13, 33642, 57524, 19842, 7724, 3400, 1072, 352, 130, 44, 14

Offset: 1

Author

Clark Kimberling, Jun 07 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....20...68
  3...10...34...116..400
  4...12...40...138..478
  5...16...54...186..644
  7...24...82...284..982
		

Crossrefs

Cf. A114537, A035513, A035506.

Programs

Mathematica

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=2*Floor[n*Sqrt[3]]   (* 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}]]
(* A191542 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191542 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)