A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 17, 18, 21, 22, 19, 16, 25, 26, 31, 32, 28, 24, 20, 36, 38, 45, 46, 41, 35, 29, 23, 52, 55, 65, 66, 59, 50, 42, 34, 27, 75, 79, 93, 94, 84, 72, 60, 49, 39, 30, 107, 113, 133, 134, 120, 103, 86, 70, 56, 43, 33, 152, 161, 189, 191, 171, 147, 123, 100, 80, 62, 48, 37

Offset: 1

Author

Clark Kimberling, Jun 03 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...7...11
  3...5...12..18..18
  6...9...14..21..31
  10..15..22..32..46
  13..19..28..41..59
		

Crossrefs

Cf. A114537, A035513, A035506.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 3/2] (* f(n) is 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]];  (* the array T *)
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191430 array *)
Flatten[Table[
  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191430 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)