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

%I #7 Mar 31 2012 20:25:45

%S 1,1,3,12,117,2250,107352,12298500,3613136949,2742962912055,

%T 5503085134707267,29497134965411187747,427365985177386403469028,

%U 16883252883454411208147060304,1832920589508888783152391724736550

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

%H Alois P. Heinz, <a href="/A133987/b133987.txt">Table of n, a(n) for n = 0..40</a>

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

%e Let b(n) = A005704(n) = number of partitions of 3n into powers of 3, then

%e the initial terms of this sequence begin:

%e b(0), b(0), b(2), b(6), b(20), b(60), b(182), b(546), b(1640),...

%e APPLICATION: SPECIAL TERNARY TREE.

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

%e Start at generation 0 with a single root node labeled [1].

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

%e labels congruent to 1(mod 3) for even n, or 3(mod 3) for odd n,

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

%e The initial generations 0..4 of the tree are as follows;

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

%e GEN.0: [1];

%e GEN.1: 1->[3];

%e GEN.2: 1-3->[1,4,7];

%e GEN.3:

%e 1-3-1->[3]

%e 1-3-4->[3,6,9,12]

%e 1-3-7->[3,6,9,12,15,18,21];

%e GEN.4:

%e 1-3-1-3->[1,4,7]

%e 1-3-4-3->[1,4,7]

%e 1-3-4-6->[1,4,7,10,13,16]

%e 1-3-4-9->[1,4,7,10,13,16,19,22,25]

%e 1-3-4-12->[1,4,7,10,13,16,19,22,25,28,31,34]

%e 1-3-7-3->[1,4,7]

%e 1-3-7-6->[1,4,7,10,13,16]

%e 1-3-7-9->[1,4,7,10,13,16,19,22,25]

%e 1-3-7-12->[1,4,7,10,13,16,19,22,25,28,31,34]

%e 1-3-7-15->[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43]

%e 1-3-7-18->[1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52]

%e 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] .

%e Note: the sum of the labels in generation n equals a(n+1) and

%e the largest term in generation n = (3^(n+1) + (-1)^(n+1) - 2)/4 + 1.

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

%Y Cf. A005704; variants: A132843, A132880.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 01 2007