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A191445 Dispersion of ([(n+1)*sqrt(3)]), where [ ]=floor, by antidiagonals.

%I A191445 #12 Oct 21 2024 00:49:46
%S A191445 1,3,2,6,5,4,12,10,8,7,22,19,15,13,9,39,34,27,24,17,11,69,60,48,43,31,
%T A191445 20,14,121,105,84,76,55,36,25,16,211,183,147,133,96,64,45,29,18,367,
%U A191445 318,256,232,168,112,79,51,32,21,637,552,445,403,292,195,138
%N A191445 Dispersion of ([(n+1)*sqrt(3)]), where [ ]=floor, by antidiagonals.
%C A191445 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 A191445 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191445 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191445 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191445 More recent examples of dispersions: A191426-A191455.
%e A191445 Northwest corner:
%e A191445   1...3...6...12..22
%e A191445   2...5...10..19..34
%e A191445   4...8...15..27..48
%e A191445   7...13..24..43..76
%e A191445   9...17..31..55..96
%t A191445 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191445 r=40; r1=12; c=40; c1=12;  x = Sqr[3];
%t A191445 f[n_] := Floor[n*x+x] (* complement of column 1 *)
%t A191445 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191445 rows = {NestList[f, 1, c]};
%t A191445 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191445 t[i_, j_] := rows[[i, j]];
%t A191445 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191445 (* A191445 array *)
%t A191445 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191445 sequence *)
%t A191445 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191445 Cf. A114537, A035513, A035506, A191442.
%K A191445 nonn,tabl
%O A191445 1,2
%A A191445 _Clark Kimberling_, Jun 04 2011