A191447 Dispersion of (1+[n*sqrt(5)]), where [ ]=floor, by antidiagonals.

1, 3, 2, 7, 5, 4, 16, 12, 9, 6, 36, 27, 21, 14, 8, 81, 61, 47, 32, 18, 10, 182, 137, 106, 72, 41, 23, 11, 407, 307, 238, 161, 92, 52, 25, 13, 911, 687, 533, 361, 206, 117, 56, 30, 15, 2038, 1537, 1192, 808, 461, 262, 126, 68, 34, 17, 4558, 3437, 2666, 1807

Offset

1,2

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.

Links

Table of n, a(n) for n=1..59.

Example

Northwest corner:
1...3....7....16...36
2...5....12...27...61
4...9....21...47...106
6...14...32...72...161
8...18...41...92...206

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_] := 1+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}]]
(* A191447 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191447 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

Crossrefs

Cf. A114537, A035513, A035506, A191446.

Sequence in context: A066884 A171429 A191448 * A191538 A361876 A080649

Adjacent sequences: A191444 A191445 A191446 * A191448 A191449 A191450

Keyword

nonn,tabl

Author

Clark Kimberling, Jun 05 2011

Status

approved