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A191443 Dispersion of the sequence ([n*sqrt(3)+1]), where [ ]=floor, read by antidiagonals.

%I A191443 #17 Oct 21 2024 00:49:54
%S A191443 1,2,3,4,6,5,7,11,9,8,13,20,16,14,10,23,35,28,25,18,12,40,61,49,44,32,
%T A191443 21,15,70,106,85,77,56,37,26,17,122,184,148,134,97,65,46,30,19,212,
%U A191443 319,257,233,169,113,80,52,33,22,368,553,446,404,293,196,139
%N A191443 Dispersion of the sequence ([n*sqrt(3)+1]), where [ ]=floor, read by antidiagonals.
%C A191443 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 A191443 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191443 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191443 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191443 More recent examples of dispersions: A191426-A191455.
%e A191443 Northwest corner:
%e A191443   1....2....4....7....13
%e A191443   3....6....11...20...35
%e A191443   5....9....16...28...49
%e A191443   8....14...25...44...77
%e A191443   10...18...32...56...97
%t A191443 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191443 r=40; r1=12; c=40; c1=12;  x = Sqrt[3];
%t A191443 f[n_] := Floor[n*x+1] (* complement of column 1 *)
%t A191443 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191443 rows = {NestList[f, 1, c]};
%t A191443 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191443 t[i_, j_] := rows[[i, j]];
%t A191443 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191443 (* A191443 array *)
%t A191443 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191443 sequence *)
%t A191443 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191443 Cf. A114537, A035513, A035506, A191442.
%K A191443 nonn,tabl
%O A191443 1,2
%A A191443 _Clark Kimberling_, Jun 04 2011