A191445 Dispersion of ([(n+1)*sqrt(3)]), where [ ]=floor, by antidiagonals.

1, 3, 2, 6, 5, 4, 12, 10, 8, 7, 22, 19, 15, 13, 9, 39, 34, 27, 24, 17, 11, 69, 60, 48, 43, 31, 20, 14, 121, 105, 84, 76, 55, 36, 25, 16, 211, 183, 147, 133, 96, 64, 45, 29, 18, 367, 318, 256, 232, 168, 112, 79, 51, 32, 21, 637, 552, 445, 403, 292, 195, 138

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...3...6...12..22
  2...5...10..19..34
  4...8...15..27..48
  7...13..24..43..76
  9...17..31..55..96
		

Crossrefs

Cf. A114537, A035513, A035506, A191442.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+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}]]
(* A191445 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191445 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)