%I #14 Oct 21 2024 01:20:18
%S 1,2,3,5,8,4,14,22,11,6,39,62,31,16,7,110,175,87,45,19,9,311,494,246,
%T 127,53,25,10,879,1397,695,359,149,70,28,12,2486,3951,1965,1015,421,
%U 197,79,33,13,7031,11175,5557,2870,1190,557,223,93,36,15,19886,31607
%N Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.
%C 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:
%C (1) s=A000040 (the primes), D=A114537, u=A114538.
%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H G. C. Greubel, <a href="/A191540/b191540.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e Northwest corner:
%e 1, 2, 5, 14, 39, ...
%e 3, 8, 22, 62, 175, ...
%e 4, 11, 31, 87, 246, ...
%e 6, 16, 45, 127, 359, ...
%e 7, 19, 53, 149, 421, ...
%t (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t r=40; r1=12; c=40; c1=12; f[n_] :=Floor[2n*Sqrt[2]] (* complement of column 1 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A191540 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191540 sequence *)
%Y Cf. A114537, A035513, A035506.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 06 2011