A191537 Dispersion of (4*n-floor(n*sqrt(2))), by antidiagonals.

1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 55, 42, 29, 13, 7, 143, 109, 75, 34, 19, 9, 370, 282, 194, 88, 50, 24, 10, 957, 730, 502, 228, 130, 63, 26, 12, 2475, 1888, 1299, 590, 337, 163, 68, 32, 14, 6400, 4882, 3359, 1526, 872, 422, 176, 83, 37, 15, 16550, 12624

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,  8,  21,  55, ...
  2,  6, 16,  42, 109, ...
  4, 11, 29,  75, 194, ...
  5, 13, 34,  88, 228, ...
  7, 19, 50, 130, 337, ...
		

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_] :=4n-Floor[n*Sqrt[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, r1}, {j, 1, c1}]]  (* A191537 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191537 sequence *)
(* Clark Kimberling, Jun 06 2011 *)