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A191444 Dispersion of ([n*sqrt(3)+3/2]), where [ ]=floor, by antidiagonals.

%I A191444 #12 Oct 21 2024 00:49:50
%S A191444 1,3,2,6,4,5,11,8,10,7,20,15,18,13,9,36,27,32,24,17,12,63,48,56,43,30,
%T A191444 22,14,110,84,98,75,53,39,25,16,192,146,171,131,93,69,44,29,19,334,
%U A191444 254,297,228,162,121,77,51,34,21,580,441,515,396,282,211,134
%N A191444 Dispersion of ([n*sqrt(3)+3/2]), where [ ]=floor, by antidiagonals.
%C A191444 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 A191444 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191444 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191444 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191444 More recent examples of dispersions: A191426-A191455.
%e A191444 Northwest corner:
%e A191444   1....3....6....11...20
%e A191444   2....4....8....15...27
%e A191444   5....10...18...32...56
%e A191444   7....13...24...43...75
%e A191444   9....17...30...53...93
%t A191444 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191444 r=40; r1=12; c=40; c1=12;  x = Sqr[3];
%t A191444 f[n_] := Floor[n*x+3/2] (* complement of column 1 *)
%t A191444 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191444 rows = {NestList[f, 1, c]};
%t A191444 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191444 t[i_, j_] := rows[[i, j]];
%t A191444 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191444 (* A191444 array *)
%t A191444 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191444 sequence *)
%t A191444 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191444 Cf. A114537, A035513, A035506, A191442.
%K A191444 nonn,tabl
%O A191444 1,2
%A A191444 _Clark Kimberling_, Jun 04 2011