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A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

%I A191438 #12 Oct 21 2024 00:47:02
%S A191438 1,2,3,4,7,5,9,16,12,6,21,38,28,14,8,50,91,67,33,19,10,120,219,161,79,
%T A191438 45,24,11,289,528,388,190,108,57,26,13,697,1274,936,458,260,137,62,31,
%U A191438 15,1682,3075,2259,1105,627,330,149,74,36,17,4060,7423,5453
%N A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.
%C A191438 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 A191438 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191438 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191438 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191438 More recent examples of dispersions: A191426-A191455.
%e A191438 Northwest corner:
%e A191438   1....2....4....9....21
%e A191438   3....7....16...38...91
%e A191438   5....12...28...67...161
%e A191438   6....14...33...79...190
%e A191438   8....19...45...108..260
%t A191438 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191438 r=40; r1=12; c=40; c1=12;  x = Sqr[2];
%t A191438 f[n_] := Floor[n*x+n] (* complement of column 1 *)
%t A191438 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191438 rows = {NestList[f, 1, c]};
%t A191438 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191438 t[i_, j_] := rows[[i, j]];
%t A191438 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191438 (* A191438 array *)
%t A191438 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191438 sequence *)
%t A191438 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191438 Cf. A114537, A035513, A035506, A191438.
%K A191438 nonn,tabl
%O A191438 1,2
%A A191438 _Clark Kimberling_, Jun 04 2011