A191448 - OEIS

%I #15 Oct 21 2024 00:51:27

%S 1,3,2,7,5,4,15,11,9,6,31,23,19,13,8,63,47,39,27,17,10,127,95,79,55,

%T 35,21,12,255,191,159,111,71,43,25,14,511,383,319,223,143,87,51,29,16,

%U 1023,767,639,447,287,175,103,59,33,18,2047,1535,1279,895,575

%N Dispersion of the odd integers greater than 1, 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.

%H Ivan Neretin, <a href="/A191448/b191448.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)

%e Northwest corner:

%e   1...3...7...15..31

%e   2...5...11..23..47

%e   4...9...19..39..79

%e   6...13..27..55..111

%e   8...17..35..71..143

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

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

%t f[n_] :=2n+1 (* 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 (* A191448 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191448 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