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A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.

%I A191442 #12 Oct 21 2024 00:49:59
%S A191442 1,2,4,3,7,6,5,12,10,8,9,21,17,14,11,16,36,29,24,19,13,28,62,50,42,33,
%T A191442 23,15,48,107,87,73,57,40,26,18,83,185,151,126,99,69,45,31,20,144,320,
%U A191442 262,218,171,120,78,54,35,22,249,554,454,378,296,208,135,94
%N A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.
%C A191442 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 A191442 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191442 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191442 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191442 More recent examples of dispersions: A191426-A191455.
%e A191442 Northwest corner:
%e A191442   1....2....3....5....9
%e A191442   4....7....12...21...36
%e A191442   6....10...17...29...50
%e A191442   8....14...24...42...73
%e A191442   11...19...33...57...99
%t A191442 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191442 r=40; r1=12; c=40; c1=12;  x = Sqr[3];
%t A191442 f[n_] := Floor[n*x+1/2] (* complement of column 1 *)
%t A191442 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191442 rows = {NestList[f, 1, c]};
%t A191442 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191442 t[i_, j_] := rows[[i, j]];
%t A191442 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191442 (* A191442 array *)
%t A191442 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191442 sequence *)
%t A191442 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191442 Cf. A114537, A035513, A035506.
%K A191442 nonn,tabl
%O A191442 1,2
%A A191442 _Clark Kimberling_, Jun 04 2011