A191434 Dispersion of ([n*x+n+3/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

1, 4, 2, 11, 6, 3, 30, 17, 9, 5, 80, 46, 25, 14, 7, 210, 121, 66, 38, 19, 8, 551, 318, 174, 100, 51, 22, 10, 1444, 834, 457, 263, 135, 59, 27, 12, 3781, 2184, 1197, 690, 354, 155, 72, 32, 13, 9900, 5719, 3135, 1807, 928, 407, 189, 85, 35, 15, 25920, 14974

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....11....30...80
  2.....6....17....46...121
  3.....9....25....66...174
  5.....14...38...100...263
  7.....19...51...135...354
		

Crossrefs

Cf. A114537, A035513, A035506, A191426.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
x = 1 + GoldenRatio; f[n_] := Floor[n*x + 3/2]
(* f(n) is 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}]]
(* A191434 array *)
Flatten[Table[
  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191434 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)