A191444 - OEIS

%I #12 Oct 21 2024 00:49:50

%S 1,3,2,6,4,5,11,8,10,7,20,15,18,13,9,36,27,32,24,17,12,63,48,56,43,30,

%T 22,14,110,84,98,75,53,39,25,16,192,146,171,131,93,69,44,29,19,334,

%U 254,297,228,162,121,77,51,34,21,580,441,515,396,282,211,134

%N Dispersion of ([n*sqrt(3)+3/2]), where [ ]=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...20

%e   2....4....8....15...27

%e   5....10...18...32...56

%e   7....13...24...43...75

%e   9....17...30...53...93

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

%t r=40; r1=12; c=40; c1=12;  x = Sqr[3];

%t f[n_] := Floor[n*x+3/2] (* 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 (* A191444 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191444 sequence *)

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

%Y Cf. A114537, A035513, A035506, A191442.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 04 2011