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A191542 Dispersion of (2*floor(n*sqrt(3))), by antidiagonals.

%I A191542 #11 Oct 21 2024 01:20:11
%S A191542 1,2,3,6,10,4,20,34,12,5,68,116,40,16,7,234,400,138,54,24,8,810,1384,
%T A191542 478,186,82,26,9,2804,4794,1654,644,284,90,30,11,9712,16606,5728,2230,
%U A191542 982,310,102,38,13,33642,57524,19842,7724,3400,1072,352,130,44,14
%N A191542 Dispersion of (2*floor(n*sqrt(3))), by antidiagonals.
%C A191542 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 A191542 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191542 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191542 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191542 More recent examples of dispersions: A191426-A191455.
%e A191542 Northwest corner:
%e A191542   1...2....6....20...68
%e A191542   3...10...34...116..400
%e A191542   4...12...40...138..478
%e A191542   5...16...54...186..644
%e A191542   7...24...82...284..982
%t A191542 (* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t A191542 r=40; r1=12; c=40; c1=12; f[n_] :=2*Floor[n*Sqrt[3]]   (* complement of column 1 *)
%t A191542 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191542 rows = {NestList[f, 1, c]};
%t A191542 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191542 t[i_, j_] := rows[[i, j]];
%t A191542 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191542 (* A191542 array *)
%t A191542 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191542 sequence *)
%t A191542 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191542 Cf. A114537, A035513, A035506.
%K A191542 nonn,tabl
%O A191542 1,2
%A A191542 _Clark Kimberling_, Jun 07 2011