A191435 Dispersion of ([n*x+n+x]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

1, 5, 2, 15, 7, 3, 41, 20, 10, 4, 109, 54, 28, 13, 6, 287, 143, 75, 36, 18, 8, 753, 376, 198, 96, 49, 23, 9, 1973, 986, 520, 253, 130, 62, 26, 11, 5167, 2583, 1363, 664, 342, 164, 70, 31, 12, 13529, 6764, 3570, 1740, 897, 431, 185, 83, 34, 14, 35421, 17710

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....5....15...41...109
  2....7....20...54...143
  3....10...28...75...198
  4....13...36...96...253
  6....18...49...130..342
		

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 = 1 + GoldenRatio;
f[n_] := Floor[n*x + x] (* f(n),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}]]
(* A191435 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191435 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)