A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

1, 4, 2, 12, 7, 3, 31, 19, 9, 5, 77, 48, 24, 14, 6, 188, 118, 60, 36, 16, 8, 456, 287, 147, 89, 41, 21, 10, 1103, 695, 357, 217, 101, 53, 26, 11, 2665, 1680, 864, 526, 246, 130, 65, 28, 13, 6436, 4058, 2088, 1272, 596, 316, 159, 70, 33, 15, 15540, 9799, 5043

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....12...31...77
  2....7....19...48...118
  3....9....24...60...147
  5....14...36...89...217
  6....16...41...101..246
		

Crossrefs

Cf. A114537, A035513, A035506, A191438.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n+x] (* 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}]]
(* A191441 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191441 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)