A141396 -id:A141396 - cp's OEIS frontend

A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

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

Offset: 1

Clark Kimberling, Jun 05 2011

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.

			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

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,...).

(* 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 *)

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

A141397 a(n) = 3*a(n-1) + A001651(n+1).

Original entry on oeis.org

1, 5, 19, 62, 193, 587, 1771, 5324, 15985, 47969, 143923, 431786, 1295377, 3886151, 11658475, 34975448, 104926369, 314779133, 944337427, 2833012310, 8499036961, 25497110915, 76491332779, 229473998372, 688421995153, 2065265985497, 6195797956531, 18587393869634

Offset: 0

Gary W. Adamson, Jun 29 2008

Row sums of triangle A141396.

			a(2) = 19 = 3*a(1) + A001651(3) = 3*5 + 4 where A001651(3) = 4.
a(2) = 19 = sum of row 2 terms of triangle A141396: (4 + 6 + 9).

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3)

Cf. A141396, A001651.

LinearRecurrence[{4,-2,-4,3},{1,5,19,62},50] (* Harvey P. Dale, Jul 07 2024 *)

Vec((-1-x-x^2) / ((1+x)*(3*x-1)*(x-1)^2) + O(x^40)) \\ Michel Marcus, Jan 21 2015

G.f.: ( -1-x-x^2 ) / ( (1+x)*(3*x-1)*(x-1)^2 ). a(n) = (-1)^n/16 -3*n/4 -3/2 +13*3^(n+1)/16. - R. J. Mathar, Feb 16 2011

cp's OEIS Frontend

A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

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