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A191448
Dispersion of the odd integers greater than 1, by antidiagonals.
6
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, 35, 21, 12, 255, 191, 159, 111, 71, 43, 25, 14, 511, 383, 319, 223, 143, 87, 51, 29, 16, 1023, 767, 639, 447, 287, 175, 103, 59, 33, 18, 2047, 1535, 1279, 895, 575
OFFSET
1,2
COMMENTS
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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)
EXAMPLE
Northwest corner:
1...3...7...15..31
2...5...11..23..47
4...9...19..39..79
6...13..27..55..111
8...17..35..71..143
MATHEMATICA
(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=2n+1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191448 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191448 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 05 2011
STATUS
approved