A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

1, 2, 3, 4, 6, 5, 8, 13, 11, 7, 17, 29, 24, 15, 9, 38, 64, 53, 33, 20, 10, 84, 143, 118, 73, 44, 22, 12, 187, 319, 263, 163, 98, 49, 26, 14, 418, 713, 588, 364, 219, 109, 58, 31, 16, 934, 1594, 1314, 813, 489, 243, 129, 69, 35, 18, 2088, 3564, 2938, 1817

Offset: 1

Author

Clark Kimberling, Jun 05 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...2....4....8...17
  3...6....13...29..64
  5...11...24...53..118
  7...15...33...73..163
  9...20...44...98..219
		

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 = Sqrt[5];
f[n_] := Floor[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}]]
(* A191446 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191446 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

Extensions

Corrected typo in name and fixed Mathematica program by Vaclav Kotesovec, Oct 24 2014