A191454 - OEIS

%I #9 Feb 13 2014 13:25:49

%S 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 160,70,28,10,1940,2550,1274,516,226,90,32,11,6276,8250,4122,1668,730,

%U 290,102,34,13,20308,26696,13338,5396,2362,938,330,110,42,14,65718

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

%C 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 (1) s=A000040 (the primes), D=A114537, u=A114538.

%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

%C More recent examples of dispersions: A191426-A191455.

%e Northwest corner:

%e 1...2....6....18...58

%e 3...8....24...76...244

%e 4...12...38...122..394

%e 5...16...50...160..516

%e 7...22...70...226..730

%t (* Program generates the dispersion array T of increasing sequence f[n] *)

%t r=40; r1=12; c=40; c1=12; x=GoldenRatio;

%t f[n_] :=2Floor[n*x]   (* complement of column 1 *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

%t rows = {NestList[f, 1, c]};

%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]

%t (* A191454 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191454 sequence *)

%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)

%Y Cf. A114537, A035513, A035506.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 05 2011