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%I #13 Sep 07 2023 07:15:53
%S 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,
%T 46,22,11,408,577,338,198,111,53,27,13,985,1393,816,478,268,128,65,31,
%U 15,2378,3363,1970,1154,647,309,157,75,36,16,5741,8119,4756
%N Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, 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.
%H Robbert Fokkink, <a href="https://arxiv.org/abs/2309.01644">The Pell Tower and Ostronometry</a>, arXiv:2309.01644 [math.CO], 2023.
%e Northwest corner:
%e 1....2....5....12...29
%e 3....7....17...41...99
%e 4....10...24...58...140
%e 6....14...34...82...198
%e 8....19...46...111..268
%t (* Program generates the dispersion array T of increasing sequence f[n] *)
%t r=40; r1=12; c=40; c1=12; x = Sqr[2];
%t f[n_] := Floor[n*x+n+1/2] (* 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, 10}, {j, 1, 10}]]
%t (* A191439 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191439 sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y Cf. A114537, A035513, A035506, A191438.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 04 2011
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