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Dispersion of (4*n-floor(n*sqrt(3))), by antidiagonals.
1

%I #13 Oct 21 2024 00:57:39

%S 1,3,2,7,5,4,16,12,10,6,37,28,23,14,8,84,64,53,32,19,9,191,146,121,73,

%T 44,21,11,434,332,275,166,100,48,25,13,985,753,624,377,227,109,57,30,

%U 15,2234,1708,1416,856,515,248,130,69,35,17,5067,3874,3212,1942

%N Dispersion of (4*n-floor(n*sqrt(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 and A191536-A191545.

%H G. C. Greubel, <a href="/A191538/b191538.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%e Northwest corner:

%e 1, 3, 7, 16, 37, ...

%e 2, 5, 12, 28, 64, ...

%e 4, 10, 23, 53, 121, ...

%e 6, 14, 32, 73, 166, ...

%e 8, 19, 44, 100, 227, ...

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

%t r=40; r1=12; c=40; c1=12; f[n_] :=4n-Floor[n*Sqrt[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, r1}, {j, 1, c1}]] (* A191538 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191538 sequence *)

%Y Cf. A114537, A035513, A035506.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 06 2011