A171370 Sequence generated from Lim:_{n..inf.} M^n, M = an infinite lower triangular matrix with (1,3,3,3,...) in every column, shifted down twice.

1, 3, 6, 12, 18, 30, 42, 66, 84, 120, 150, 210, 252, 336, 402, 534, 618, 786, 906, 1146, 1296, 1596, 1806, 2226, 2478, 2982, 3318, 3990, 4392, 5196, 5730, 6798, 7416, 8652, 9438, 11010, 11916, 13728, 14874, 17166, 18462, 21054, 22650, 25842, 27648, 31260

Offset: 0

Author

Gary W. Adamson, Dec 06 2009

Keywords

• nonn

Comments

A000123 can be generated through an analogous procedure replacing (1,3,3,3,...) with (1,2,2,2,...).

A171370 has the property that (1, 3, 6, 12, 18,...) / (1, 3, 3, 3,..) generates an aerated variant: (1, 0, 3, 0, 6, 0, 12,...).

Similarly, given A000123; (1, 2, 4, 6, 10, 14,...) / (1, 2, 2, 2,...) generates an aerated variant: (1, 0, 2, 0, 6, 0, 10,...).

Row sums of the generating triangle = A032766 starting with 1. - Gary W. Adamson, Feb 15 2010

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000123.

Cf. A032766. - Gary W. Adamson, Feb 15 2010

Programs

Maple

a:= n-> (Matrix(n+1, (i, j)-> `if`(i=2*j-1, 1,
        `if`(i>2*j-1, 3, 0)))^n)[n+1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 16 2014

Formula

Let M = an infinite lower triangular matrix with (1,3,3,3,...) in every column shifted down twice:

1;

3;

3, 1;

3, 3;

3, 3, 1;

3, 3, 3;

...

Sequence A171370 = Lim:_{n..inf.} M^n, the left-shifted vector considered as a sequence.

Extensions

a(20)-a(45) from Alois P. Heinz, Apr 16 2014