

A191449


Dispersion of (3,6,9,12,15,...), by antidiagonals.


7



1, 3, 2, 9, 6, 4, 27, 18, 12, 5, 81, 54, 36, 15, 7, 243, 162, 108, 45, 21, 8, 729, 486, 324, 135, 63, 24, 10, 2187, 1458, 972, 405, 189, 72, 30, 11, 6561, 4374, 2916, 1215, 567, 216, 90, 33, 13, 19683, 13122, 8748, 3645, 1701, 648, 270, 99, 39, 14, 59049
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OFFSET

1,2


COMMENTS

Transpose of A141396.
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 nth 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: A191426A191455.


LINKS

Table of n, a(n) for n=1..56.


FORMULA

T(i,j)=T(i,1)*T(1,j)=floor((3i1)/2)*3^(j1).


EXAMPLE

Northwest corner:
1...3....9....27...81
2...6....18...54...162
4...12...36...108..324
5...15...45...135..405
7...21...63...189..567


MATHEMATICA

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n (* 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}]]
(* A191449 array *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]] (* A191449 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)


CROSSREFS

Cf. A114537, A035513, A035506,
A054582: dispersion of (2,4,6,8,...).
A191450: dispersion of (2,5,8,11,...).
A191451: dispersion of (4,7,10,13,...).
A191452: dispersion of (4,8,12,16,...).
Sequence in context: A289053 A191539 A235539 * A175840 A125152 A229119
Adjacent sequences: A191446 A191447 A191448 * A191450 A191451 A191452


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 05 2011


STATUS

approved



