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%I #13 Dec 09 2017 19:37:48
%S 1,4,2,8,5,3,14,10,7,6,22,17,12,11,9,34,27,19,18,15,13,51,41,29,28,24,
%T 21,16,75,60,44,42,36,32,25,20,109,87,65,62,53,48,38,31,23,157,126,94,
%U 90,77,70,56,46,35,26,225,181,135,130,111,101,82,68,52,39
%N Dispersion of (3+floor(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="/A191536/b191536.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e Northwest corner:
%e 1...4....8....14...22
%e 2...5....10...17...27
%e 3...7....12...19...29
%e 6...11...18...28...42
%e 9...15...24...36...54
%t (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t r=40; r1=12; c=40; c1=12; f[n_] :=3+Floor[n*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}]]
%t (* A191536 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y Cf. A114537, A035513, A035506.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 06 2011
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