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A133987 a(n) = A005704( (3^n + (-1)^n - 2)/4 ), where A005704(n) = number of partitions of 3n into powers of 3. 1
1, 1, 3, 12, 117, 2250, 107352, 12298500, 3613136949, 2742962912055, 5503085134707267, 29497134965411187747, 427365985177386403469028, 16883252883454411208147060304, 1832920589508888783152391724736550 (list; graph; refs; listen; history; text; internal format)
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: A308144 A320257 A009254 * A194506 A280458 A294198

Adjacent sequences:  A133984 A133985 A133986 * A133988 A133989 A133990

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 01 2007

STATUS

approved

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Last modified June 15 11:10 EDT 2021. Contains 345048 sequences. (Running on oeis4.)