0,3

Computed by Max Alekseyev (maxal(AT)cs.ucsd.edu), Aug 31 2007.

Max Alekseyev and Dean Hickerson, Table of n, a(n) for n = 0..26

a(n) = A000123( (2^(n+1) + (-1)^n - 3)/6 ) for n>=0. a(n) = a(n-1) + A000123( A000975(n-1) - 1) for n>0.

Let b(n) = A000123(n) = number of partitions of 2n into powers of 2, then the initial terms of this sequence begin:

b(0), b(0), b(1), b(2), b(5), b(10), b(21), b(42), b(85), b(170),...

OSCILLATING PARITY 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 k child nodes with labels having opposite parity to k within the range {1..2k}.

The initial generations 0..5 of the tree begin as follows; the path from the root node is given, followed by child nodes in [].

GEN.0: [1];

GEN.1: 1->[2];

GEN.2: 1-2->[1,3];

GEN.3:

1-2-1->[2]

1-2-3->[2,4,6]

GEN.4:

1-2-1-2->[1,3]

1-2-3-2->[1,3]

1-2-3-4->[1,3,5,7]

1-2-3-6->[1,3,5,7,9,11];

GEN.5:

1-2-1-2-1->[2]

1-2-1-2-3->[2,4,6]

1-2-3-2-1->[2]

1-2-3-2-3->[2,4,6]

1-2-3-4-1->[2]

1-2-3-4-3->[2,4,6]

1-2-3-4-5->[2,4,6,8,10]

1-2-3-4-7->[2,4,6,8,10,12,14]

1-2-3-6-1->[2]

1-2-3-6-3->[2,4,6]

1-2-3-6-5->[2,4,6,8,10]

1-2-3-6-7->[2,4,6,8,10,12,14]

1-2-3-6-9->[2,4,6,8,10,12,14,16,18]

1-2-3-6-11->[2,4,6,8,10,12,14,16,18,20,22] .

Note: largest node label in generation n is A000975(n) + 1.

(PARI) {A000123(n) = if(n<1, n==0, A000123(n\2) + A000123(n-1))} {a(n) = A000123( (2^(n+1) + (-1)^n - 3)/6 )}

Cf. A000123, A000975.

Sequence in context: A030808 A047009 A027740 this_sequence A019537 A046911 A089127

Adjacent sequences: A132877 A132878 A132879 this_sequence A132881 A132882 A132883

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 11 2007