A191427 - OEIS

%I #17 Oct 20 2024 21:02:09

%S 1,3,2,6,4,5,11,7,9,8,19,12,16,14,10,32,20,27,24,17,13,53,33,45,40,29,

%T 22,15,87,54,74,66,48,37,25,18,142,88,121,108,79,61,41,30,21,231,143,

%U 197,176,129,100,67,50,35,23,375,232,320,286,210,163,109,82,58,38,26,608,376,519,464,341,265,177,134,95,62,43,28

%N Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 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...3...6...11..19

%e   2...4...7...12..20

%e   5...9...16..27..45

%e   8...14..24..40..66

%e   10..17..29..48..79

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

%t r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)

%t c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)

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

%t mex[list_] :=

%t 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 (* A191427 array *)

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

%t (* A191427 sequence *)

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

%Y Cf. A114537, A035513, A035506.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 02 2011