A191439 Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, by antidiagonals.

1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 29, 41, 24, 14, 8, 70, 99, 58, 34, 19, 9, 169, 239, 140, 82, 46, 22, 11, 408, 577, 338, 198, 111, 53, 27, 13, 985, 1393, 816, 478, 268, 128, 65, 31, 15, 2378, 3363, 1970, 1154, 647, 309, 157, 75, 36, 16, 5741, 8119, 4756

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..58.

Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.

Example

Northwest corner:
1....2....5....12...29
3....7....17...41...99
4....10...24...58...140
6....14...34...82...198
8....19...46...111..268

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+1/2] (* 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}]]
(* A191439 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191439 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

Crossrefs

Cf. A114537, A035513, A035506, A191438.

Sequence in context: A369515 A103866 A268711 * A191723 A292874 A302847

Adjacent sequences: A191436 A191437 A191438 * A191440 A191441 A191442

Keyword

nonn,tabl

Author

Clark Kimberling, Jun 04 2011

Status

approved