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A191540 Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.

%I A191540 #14 Oct 21 2024 01:20:18
%S A191540 1,2,3,5,8,4,14,22,11,6,39,62,31,16,7,110,175,87,45,19,9,311,494,246,
%T A191540 127,53,25,10,879,1397,695,359,149,70,28,12,2486,3951,1965,1015,421,
%U A191540 197,79,33,13,7031,11175,5557,2870,1190,557,223,93,36,15,19886,31607
%N A191540 Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.
%C A191540 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 A191540 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191540 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191540 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191540 More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H A191540 G. C. Greubel, <a href="/A191540/b191540.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A191540 Northwest corner:
%e A191540   1,  2,  5,  14,  39, ...
%e A191540   3,  8, 22,  62, 175, ...
%e A191540   4, 11, 31,  87, 246, ...
%e A191540   6, 16, 45, 127, 359, ...
%e A191540   7, 19, 53, 149, 421, ...
%t A191540 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191540 r=40; r1=12; c=40; c1=12; f[n_] :=Floor[2n*Sqrt[2]]   (* complement of column 1 *)
%t A191540 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191540 rows = {NestList[f, 1, c]};
%t A191540 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191540 t[i_, j_] := rows[[i, j]];
%t A191540 TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A191540 array *)
%t A191540 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191540 sequence *)
%Y A191540 Cf. A114537, A035513, A035506.
%K A191540 nonn,tabl
%O A191540 1,2
%A A191540 _Clark Kimberling_, Jun 06 2011