A191539 Dispersion of (5*n-floor(n*sqrt(5))), by antidiagonals.

1, 3, 2, 9, 6, 4, 25, 17, 12, 5, 70, 47, 34, 14, 7, 194, 130, 94, 39, 20, 8, 537, 360, 260, 108, 56, 23, 10, 1485, 996, 719, 299, 155, 64, 28, 11, 4105, 2753, 1988, 827, 429, 177, 78, 31, 13, 11346, 7610, 5495, 2286, 1186, 490, 216, 86, 36, 15, 31360, 21034

Offset: 1

Author

Clark Kimberling, Jun 06 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 and A191536-A191545.

Examples

			Northwest corner:
  1...3....9....25...70
  2...6....17...47...130
  4...12...34...94...260
  5...14...39...108..299
  7...20...56...155..429
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A114537, A035513, A035506.

Programs

Mathematica

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=5n-Floor[n*Sqrt[5]]   (* 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, r1}, {j, 1, c1}]]  (* A191539 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191539 sequence *)