A191429 - OEIS

%I #13 Oct 20 2024 21:01:39

%S 1,3,2,6,4,5,10,7,9,8,16,11,14,13,12,24,17,21,20,18,15,35,26,31,30,27,

%T 23,19,51,38,45,44,40,34,28,22,74,55,65,64,58,50,41,33,25,106,79,93,

%U 92,84,72,59,48,37,29,151,113,133,132,120,103,85,69,54,43,32,215,161,190,188,171,147,122,99,78,62,47,36

%N Dispersion of ([n*sqrt(2)+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...10..16

%e   2...4...7...11..17

%e   5...9...14..21..31

%e   8...13..20..30..44

%e   12..18..27..40..58

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

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

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

%t x = Sqrt[2];

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

%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]];  (* the array T *)

%t TableForm[

%t Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191429 array *)

%t Flatten[Table[

%t   t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191429 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 03 2011