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A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

%I A191440 #12 Oct 21 2024 00:46:51
%S A191440 1,3,2,8,6,4,20,15,11,5,49,37,28,13,7,119,90,69,32,18,9,288,218,168,
%T A191440 78,44,23,10,696,527,407,189,107,57,25,12,1681,1273,984,457,259,139,
%U A191440 61,30,14,4059,3074,2377,1104,626,337,148,73,35,16,9800,7422,5740
%N A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.
%C A191440 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 A191440 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191440 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191440 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191440 More recent examples of dispersions: A191426-A191455.
%e A191440 Northwest corner:
%e A191440   1....3....8....20...49
%e A191440   2....6....15...37...90
%e A191440   4....11...28...69...168
%e A191440   5....13...32...78...189
%e A191440   7....18...44...107..259
%t A191440 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191440 r=40; r1=12; c=40; c1=12;  x = Sqr[2];
%t A191440 f[n_] := Floor[n*x+n+3/2] (* complement of column 1 *)
%t A191440 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191440 rows = {NestList[f, 1, c]};
%t A191440 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191440 t[i_, j_] := rows[[i, j]];
%t A191440 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191440 (* A191440 array *)
%t A191440 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191440 sequence *)
%t A191440 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191440 Cf. A114537, A035513, A035506, A191438, A191439, A083087.
%K A191440 nonn,tabl
%O A191440 1,2
%A A191440 _Clark Kimberling_, Jun 04 2011