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A191544 Dispersion of (floor(7n/3)), by antidiagonals. 1

%I

%S 1,2,3,4,7,5,9,16,11,6,21,37,25,14,8,49,86,58,32,18,10,114,200,135,74,

%T 42,23,12,266,466,315,172,98,53,28,13,620,1087,735,401,228,123,65,30,

%U 15,1446,2536,1715,935,532,287,151,70,35,17,3374,5917,4001,2181

%N Dispersion of (floor(7n/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....4....9...21

%e 3...7....16...37..86

%e 5...11...25...58..135

%e 6...14...32...74..172

%e 8...18...42...98..228

%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[7n/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 (* A191544 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191544 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 09 2011

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Last modified April 1 04:15 EDT 2020. Contains 333155 sequences. (Running on oeis4.)