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A191454 Dispersion of (2*floor(n*r)), where r=(golden ratio), by antidiagonals.

%I A191454 #12 Oct 21 2024 00:55:48
%S A191454 1,2,3,6,8,4,18,24,12,5,58,76,38,16,7,186,244,122,50,22,9,600,788,394,
%T A191454 160,70,28,10,1940,2550,1274,516,226,90,32,11,6276,8250,4122,1668,730,
%U A191454 290,102,34,13,20308,26696,13338,5396,2362,938,330,110,42,14,65718
%N A191454 Dispersion of (2*floor(n*r)), where r=(golden ratio), by antidiagonals.
%C A191454 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 A191454 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191454 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191454 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191454 More recent examples of dispersions: A191426-A191455.
%e A191454 Northwest corner:
%e A191454   1...2....6....18...58
%e A191454   3...8....24...76...244
%e A191454   4...12...38...122..394
%e A191454   5...16...50...160..516
%e A191454   7...22...70...226..730
%t A191454 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191454 r=40; r1=12; c=40; c1=12; x=GoldenRatio;
%t A191454 f[n_] :=2Floor[n*x]   (* complement of column 1 *)
%t A191454 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191454 rows = {NestList[f, 1, c]};
%t A191454 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191454 t[i_, j_] := rows[[i, j]];
%t A191454 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191454 (* A191454 array *)
%t A191454 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191454 sequence *)
%t A191454 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191454 Cf. A114537, A035513, A035506.
%K A191454 nonn,tabl
%O A191454 1,2
%A A191454 _Clark Kimberling_, Jun 05 2011