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%I #9 Feb 14 2014 00:29:32
%S 1,4,2,12,6,3,33,17,9,5,88,46,25,14,7,232,122,67,38,19,8,609,321,177,
%T 101,51,22,10,1596,842,465,266,135,59,27,11,4180,2206,1219,698,355,
%U 156,72,30,13,10945,5777,3193,1829,931,410,190,80,35,15,28656,15126
%N Dispersion of ([nx+n+x-1]), where x=(golden ratio) and [ ]=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.
%e Northwest corner:
%e 1.....4....12...33...88
%e 2.....6....17...46...122
%e 3.....9....25...67...177
%e 5.....14...38...101..266
%e 7.....19...51...135..355
%t (* Program generates the dispersion array T of increasing sequence f[n] *)
%t r = 40; r1 = 12; c = 40; c1 = 12; x = GoldenRatio;
%t f[n_] := Floor[n*x+n+x-1] (* 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 (* A191436 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191436 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 04 2011
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