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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
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 01 2007
STATUS
approved