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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 (list; table; graph; refs; listen; history; text; internal format)
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 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

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

FORMULA

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

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

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Last modified December 8 02:32 EST 2021. Contains 349590 sequences. (Running on oeis4.)