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A191429 Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals.

%I A191429 #13 Oct 20 2024 21:01:39
%S A191429 1,3,2,6,4,5,10,7,9,8,16,11,14,13,12,24,17,21,20,18,15,35,26,31,30,27,
%T A191429 23,19,51,38,45,44,40,34,28,22,74,55,65,64,58,50,41,33,25,106,79,93,
%U A191429 92,84,72,59,48,37,29,151,113,133,132,120,103,85,69,54,43,32,215,161,190,188,171,147,122,99,78,62,47,36
%N A191429 Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals.
%C A191429 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 A191429 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191429 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191429 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191429 More recent examples of dispersions: A191426-A191455.
%e A191429 Northwest corner:
%e A191429   1...3...6...10..16
%e A191429   2...4...7...11..17
%e A191429   5...9...14..21..31
%e A191429   8...13..20..30..44
%e A191429   12..18..27..40..58
%t A191429 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191429 r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
%t A191429 c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
%t A191429 x = Sqrt[2];
%t A191429 f[n_] := Floor[n*x + 2] (* f(n) is complement of column 1 *)
%t A191429 mex[list_] :=
%t A191429 NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191429   Length[Union[list]]]
%t A191429 rows = {NestList[f, 1, c]};
%t A191429 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191429 t[i_, j_] := rows[[i, j]];  (* the array T *)
%t A191429 TableForm[
%t A191429 Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191429 array *)
%t A191429 Flatten[Table[
%t A191429   t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191429 sequence *)
%t A191429 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191429 Cf. A114537, A035513, A035506.
%K A191429 nonn,tabl
%O A191429 1,2
%A A191429 _Clark Kimberling_, Jun 03 2011