0,2

a(n) = A005704( (5*3^n + (-1)^n - 6)/8 ).

Let b(n) = A000123(n) = number of partitions of 3n into powers of 3,

then the initial terms of this sequence begin:

b(0), b(1), b(5), b(16), b(50), b(151), b(455), b(1366),...

APPLICATION: SPECIAL TERNARY TREE.

a(n) = number of nodes in generation n of the following tree.

Start at generation 0 with a single root node labeled [2].

From then on, each parent node [k] is attached k child nodes with

labels congruent to 2(mod 3) for even n, or 3(mod 3) for odd n,

within the range {1..3k}, for generation n >= 0.

The initial generations 0..3 of the tree begin as follows;

the path from the root node is given, followed by child nodes in [].

GEN.0: [2];

GEN.1: 2->[3,6];

GEN.2:

2-3->[2,5,8]

2-6->[2,5,8,11,14,17];

GEN.3:

2-3-2->[3,6]

2-3-5->[3,6,9,12,15]

2-3-8->[3,6,9,12,15,18,21,24]

2-6-2->[3,6]

2-6-5->[3,6,9,12,15]

2-6-8->[3,6,9,12,15,18,21,24]

2-6-11->[3,6,9,12,15,18,21,24,27,30,33]

2-6-14->[3,6,9,12,15,18,21,24,27,30,33,36,39,42]

2-6-17->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51] .

Note: largest node label in generation n is A037480(n) + 1,

and the sum of the labels in generation n equals a(n+1).

(PARI) {A005704(n) = if(n<1, n==0, A005704(n\3) + A005704(n-1))} {a(n) = A005704( (5*3^n + (-1)^n - 6)/8 )}

Cf. A005704, A037480; variant: A132880.

Sequence in context: A038035 A133984 A108995 this_sequence A084873 A103996 A015473

Adjacent sequences: A132840 A132841 A132842 this_sequence A132844 A132845 A132846

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 27 2007