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A191537 Dispersion of (4*n-floor(n*sqrt(2))), by antidiagonals.

%I A191537 #20 Oct 21 2024 00:55:37
%S A191537 1,3,2,8,6,4,21,16,11,5,55,42,29,13,7,143,109,75,34,19,9,370,282,194,
%T A191537 88,50,24,10,957,730,502,228,130,63,26,12,2475,1888,1299,590,337,163,
%U A191537 68,32,14,6400,4882,3359,1526,872,422,176,83,37,15,16550,12624
%N A191537 Dispersion of (4*n-floor(n*sqrt(2))), by antidiagonals.
%C A191537 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 A191537 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191537 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191537 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191537 More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H A191537 G. C. Greubel, <a href="/A191537/b191537.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A191537 Northwest corner:
%e A191537   1,  3,  8,  21,  55, ...
%e A191537   2,  6, 16,  42, 109, ...
%e A191537   4, 11, 29,  75, 194, ...
%e A191537   5, 13, 34,  88, 228, ...
%e A191537   7, 19, 50, 130, 337, ...
%t A191537 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191537 r=40; r1=12; c=40; c1=12; f[n_] :=4n-Floor[n*Sqrt[2]]   (* complement of column 1 *)
%t A191537 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191537 rows = {NestList[f, 1, c]};
%t A191537 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191537 t[i_, j_] := rows[[i, j]];
%t A191537 TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A191537 array *)
%t A191537 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191537 sequence *)
%t A191537 (* _Clark Kimberling_, Jun 06 2011 *)
%Y A191537 Cf. A114537, A035513, A035506.
%K A191537 nonn,tabl
%O A191537 1,2
%A A191537 _Clark Kimberling_, Jun 06 2011