A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

1, 3, 2, 6, 4, 5, 11, 8, 9, 7, 19, 14, 16, 12, 10, 32, 24, 27, 21, 17, 13, 53, 40, 45, 35, 29, 22, 15, 87, 66, 74, 58, 48, 37, 25, 18, 142, 108, 121, 95, 79, 61, 42, 30, 20, 231, 176, 197, 155, 129, 100, 69, 50, 33, 23, 375, 286, 320, 252, 210, 163, 113, 82, 55, 38, 26, 608, 464, 519, 409, 341, 265, 184, 134, 90, 63, 43, 28

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...3...6...11..19
  2...4...8...14..24
  5...9...16..27..45
  7...12..21..35..58
  10..17..29..48..79
		

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, r1=# rows to show *)
c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + x]
(* 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]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191428 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191428 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)