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A191538 Dispersion of (4*n-floor(n*sqrt(3))), by antidiagonals.

%I A191538 #13 Oct 21 2024 00:57:39
%S A191538 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 A191538 44,21,11,434,332,275,166,100,48,25,13,985,753,624,377,227,109,57,30,
%U A191538 15,2234,1708,1416,856,515,248,130,69,35,17,5067,3874,3212,1942
%N A191538 Dispersion of (4*n-floor(n*sqrt(3))), by antidiagonals.
%C A191538 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 A191538 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191538 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191538 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191538 More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H A191538 G. C. Greubel, <a href="/A191538/b191538.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A191538 Northwest corner:
%e A191538   1,  3,  7,  16,  37, ...
%e A191538   2,  5, 12,  28,  64, ...
%e A191538   4, 10, 23,  53, 121, ...
%e A191538   6, 14, 32,  73, 166, ...
%e A191538   8, 19, 44, 100, 227, ...
%t A191538 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191538 r=40; r1=12; c=40; c1=12; f[n_] :=4n-Floor[n*Sqrt[3]]   (* complement of column 1 *)
%t A191538 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191538 rows = {NestList[f, 1, c]};
%t A191538 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191538 t[i_, j_] := rows[[i, j]];
%t A191538 TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A191538 array *)
%t A191538 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191538 sequence *)
%Y A191538 Cf. A114537, A035513, A035506.
%K A191538 nonn,tabl
%O A191538 1,2
%A A191538 _Clark Kimberling_, Jun 06 2011