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Dispersion of (floor(8*n/3)), by antidiagonals.
1

%I #11 Oct 21 2024 01:20:08

%S 1,2,3,5,8,4,13,21,10,6,34,56,26,16,7,90,149,69,42,18,9,240,397,184,

%T 112,48,24,11,640,1058,490,298,128,64,29,12,1706,2821,1306,794,341,

%U 170,77,32,14,4549,7522,3482,2117,909,453,205,85,37,15,12130,20058

%N Dispersion of (floor(8*n/3)), 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...2....5....13...34

%e 3...8....21...56...149

%e 4...10...26...69...184

%e 6...16...42...112..298

%e 7...18...48...128..341

%t (* Program generates the dispersion array T of the complement of increasing sequence f[n] *)

%t r=40; r1=12; c=40; c1=12; f[n_] :=Floor[8n/3]] (* 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 (* A191543 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191543 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 07 2011