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A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

%I A191430 #14 Oct 20 2024 21:02:38
%S A191430 1,2,3,4,5,6,7,8,9,10,11,12,14,15,13,17,18,21,22,19,16,25,26,31,32,28,
%T A191430 24,20,36,38,45,46,41,35,29,23,52,55,65,66,59,50,42,34,27,75,79,93,94,
%U A191430 84,72,60,49,39,30,107,113,133,134,120,103,86,70,56,43,33,152,161,189,191,171,147,123,100,80,62,48,37
%N A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.
%C A191430 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 A191430 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191430 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191430 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191430 More recent examples of dispersions: A191426-A191455.
%e A191430 Northwest corner:
%e A191430   1...2...4...7...11
%e A191430   3...5...12..18..18
%e A191430   6...9...14..21..31
%e A191430   10..15..22..32..46
%e A191430   13..19..28..41..59
%t A191430 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191430 r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
%t A191430 c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
%t A191430 x = Sqrt[2];
%t A191430 f[n_] := Floor[n*x + 3/2] (* f(n) is complement of column 1 *)
%t A191430 mex[list_] :=
%t A191430 NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191430   Length[Union[list]]]
%t A191430 rows = {NestList[f, 1, c]};
%t A191430 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191430 t[i_, j_] := rows[[i, j]];  (* the array T *)
%t A191430 TableForm[
%t A191430 Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191430 array *)
%t A191430 Flatten[Table[
%t A191430   t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191430 sequence *)
%t A191430 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191430 Cf. A114537, A035513, A035506.
%K A191430 nonn,tabl
%O A191430 1,2
%A A191430 _Clark Kimberling_, Jun 03 2011