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%I #9 Feb 14 2014 00:27:49
%S 1,3,2,6,4,5,11,8,9,7,19,14,16,12,10,32,24,27,21,17,13,53,40,45,35,29,
%T 22,15,87,66,74,58,48,37,25,18,142,108,121,95,79,61,42,30,20,231,176,
%U 197,155,129,100,69,50,33,23,375,286,320,252,210,163,113,82,55,38,26,608,464,519,409,341,265,184,134,90,63,43,28
%N Dispersion of ([nr+r]), where r=(golden ratio)=(1+sqrt(5))/2 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...3...6...11..19
%e 2...4...8...14..24
%e 5...9...16..27..45
%e 7...12..21..35..58
%e 10..17..29..48..79
%t (* Program generates the dispersion array T of increasing sequence f[n] *)
%t r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
%t c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
%t x = GoldenRatio; f[n_] := Floor[n*x + x]
%t (* f(n) is complement of column 1 *)
%t mex[list_] :=
%t NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t 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 (* A191428 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t (* A191428 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 03 2011
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