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
Examples
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).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..40
Programs
Formula
a(n) = A005704( (5*3^n + (-1)^n - 6)/8 ).
Comments