A133987 a(n) = A005704( (3^n + (-1)^n - 2)/4 ), where A005704(n) = number of partitions of 3n into powers of 3.

1, 1, 3, 12, 117, 2250, 107352, 12298500, 3613136949, 2742962912055, 5503085134707267, 29497134965411187747, 427365985177386403469028, 16883252883454411208147060304, 1832920589508888783152391724736550

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..40

Formula

(3^n + (-1)^n - 2)/4 gives the n-th number that has alternating base-3 digits {0,2} (starting with zero).

Example

Let b(n) = A005704(n) = number of partitions of 3n into powers of 3, then
the initial terms of this sequence begin:
b(0), b(0), b(2), b(6), b(20), b(60), b(182), b(546), b(1640),...
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 [1].
From then on, each parent node [k] is attached to k child nodes with
labels congruent to 1(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..4 of the tree are as follows;
the path from the root node is given, followed by child nodes in [].
GEN.0: [1];
GEN.1: 1->[3];
GEN.2: 1-3->[1,4,7];
GEN.3:
1-3-1->[3]
1-3-4->[3,6,9,12]
1-3-7->[3,6,9,12,15,18,21];
GEN.4:
1-3-1-3->[1,4,7]
1-3-4-3->[1,4,7]
1-3-4-6->[1,4,7,10,13,16]
1-3-4-9->[1,4,7,10,13,16,19,22,25]
1-3-4-12->[1,4,7,10,13,16,19,22,25,28,31,34]
1-3-7-3->[1,4,7]
1-3-7-6->[1,4,7,10,13,16]
1-3-7-9->[1,4,7,10,13,16,19,22,25]
1-3-7-12->[1,4,7,10,13,16,19,22,25,28,31,34]
1-3-7-15->[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43]
1-3-7-18->[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52]
1-3-7-21->[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61] .
Note: the sum of the labels in generation n equals a(n+1) and
the largest term in generation n = (3^(n+1) + (-1)^(n+1) - 2)/4 + 1.

Prog

(PARI) {A005704(n) = if(n<1, n==0, A005704(n\3) + A005704(n-1))} {a(n) = A005704( (3^n + (-1)^n - 2)/4 )}

Crossrefs

Cf. A005704; variants: A132843, A132880.

Sequence in context: A350410 A009254 A377066 * A359658 A194506 A280458

Adjacent sequences: A133984 A133985 A133986 * A133988 A133989 A133990

Keyword

nonn

Author

Paul D. Hanna, Oct 01 2007

Status

approved