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A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

%I A191452 #16 Oct 21 2024 00:52:25
%S A191452 1,4,2,16,8,3,64,32,12,5,256,128,48,20,6,1024,512,192,80,24,7,4096,
%T A191452 2048,768,320,96,28,9,16384,8192,3072,1280,384,112,36,10,65536,32768,
%U A191452 12288,5120,1536,448,144,40,11,262144,131072,49152,20480,6144,1792,576
%N A191452 Dispersion of (4,8,12,16,...), by antidiagonals.
%C A191452 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 A191452 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191452 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191452 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191452 More recent examples of dispersions: A191426-A191455.
%H A191452 Ivan Neretin, <a href="/A191452/b191452.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191452 Northwest corner:
%e A191452   1...4....16...64....256
%e A191452   2...8....32...128...512
%e A191452   3...12...48...192...768
%e A191452   5...20...80...320...1280
%e A191452   6...24...96...384...1536
%t A191452 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191452 r=40; r1=12; c=40; c1=12;
%t A191452 f[n_] :=4n  (* complement of column 1 *)
%t A191452 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191452 rows = {NestList[f, 1, c]};
%t A191452 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191452 t[i_, j_] := rows[[i, j]];
%t A191452 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191452 (* A191452 array *)
%t A191452 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191452 sequence *)
%t A191452 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191452 Cf. A114537, A035513, A035506, A191449.
%K A191452 nonn,tabl
%O A191452 1,2
%A A191452 _Clark Kimberling_, Jun 05 2011