A132880 -id:A132880 - cp's OEIS frontend

A132843 a(n) = A005704( A037480(n) ) for n>0 with a(0)=1, where A005704 = number of partitions of 3n into powers of 3 and A037480(n) = n-th number having alternating base-3 digits 1, 2 (starting with '1').

1, 2, 9, 72, 1296, 52407, 5240052, 1314516033, 853923545352, 1457086698392796, 6631460154689418828, 81384300080656595328843, 2719577128999047606509974434, 249432083657086432899494832228657

Offset: 0

Paul D. Hanna, Sep 27 2007

			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(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).

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

Cf. A005704, A037480; variant: A132880.

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

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

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Oct 01 2007

nonn

			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.

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

Cf. A005704; variants: A132843, A132880.

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

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

cp's OEIS Frontend

A132843 a(n) = A005704( A037480(n) ) for n>0 with a(0)=1, where A005704 = number of partitions of 3n into powers of 3 and A037480(n) = n-th number having alternating base-3 digits 1, 2 (starting with '1').

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