A191444 Dispersion of ([n*sqrt(3)+3/2]), where [ ]=floor, by antidiagonals.

1, 3, 2, 6, 4, 5, 11, 8, 10, 7, 20, 15, 18, 13, 9, 36, 27, 32, 24, 17, 12, 63, 48, 56, 43, 30, 22, 14, 110, 84, 98, 75, 53, 39, 25, 16, 192, 146, 171, 131, 93, 69, 44, 29, 19, 334, 254, 297, 228, 162, 121, 77, 51, 34, 21, 580, 441, 515, 396, 282, 211, 134

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....11...20
  2....4....8....15...27
  5....10...18...32...56
  7....13...24...43...75
  9....17...30...53...93
		

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+3/2] (* 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}]]
(* A191444 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191444 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)