A191434 - OEIS

%I #9 Feb 14 2014 00:29:05

%S 1,4,2,11,6,3,30,17,9,5,80,46,25,14,7,210,121,66,38,19,8,551,318,174,

%T 100,51,22,10,1444,834,457,263,135,59,27,12,3781,2184,1197,690,354,

%U 155,72,32,13,9900,5719,3135,1807,928,407,189,85,35,15,25920,14974

%N Dispersion of ([nx+n+3/2]), where x=(golden ratio) and [ ]=floor, 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.....4....11....30...80

%e 2.....6....17....46...121

%e 3.....9....25....66...174

%e 5.....14...38...100...263

%e 7.....19...51...135...354

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

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

%t x = 1 + GoldenRatio; f[n_] := Floor[n*x + 3/2]

%t (* f(n) is complement of column 1 *)

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

%t   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 (* A191434 array *)

%t Flatten[Table[

%t   t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]

%t (* A191434 sequence *)

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

%Y Cf. A114537, A035513, A035506, A191426.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 04 2011