A132880 a(n) = A000123( A000975(n-1) ) for n>=1 with a(0)=1, where A000123(n) = number of partitions of 2n into powers of 2 and A000975(n) = n-th number without consecutive equal binary digits.

1, 1, 2, 4, 14, 60, 450, 4964, 95982, 3037948, 170005730, 16522010532, 2882717916878, 902450057292988, 514768747418386946, 537142988843543963620, 1033976171696917695108270, 3688322935382700002945333884, 24514290054855626308893309022498

Offset: 0

Author

Paul D. Hanna, Sep 11 2007

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Max Alekseyev and Dean Hickerson, Table of n, a(n) for n = 0..50 (terms 0..26 by Max Alekseyev and Dean Hickerson)

Crossrefs

Cf. A000123, A000975.

Programs

Maple

g:= proc(b, n) option remember; local t; if b<0 then 0 elif b=0 or n=0 then 1 elif b>=n then add (g(b-t, n) *binomial (n+1, t) *(-1)^(t+1), t=1..n+1) else g(b-1, n) +g(2*b, n-1) fi end: a:= proc(n) local m, t; m:= ceil (2 *(2^(n-1) -1) /3); t:= ilog2(2*m+1); g(m /2^(t-1), t) end: seq (a(n), n=0..30);  # Alois P. Heinz, Apr 17 2009

Mathematica

g[b_, n_] := g[b, n] = Module[{t}, Which[b<0, 0, b == 0 || n == 0, 1, b >= n, Sum[g[b-t, n]*Binomial[n+1, t]*(-1)^(t+1), {t, 1, n+1}], True, g[b-1, n] + g[2*b, n-1]]]; a[n_] := Module[{m, t}, m = Ceiling[2*(2^(n-1)-1) /3]; t = Log[2, 2*m+1] // Floor; g[m /2^(t-1), t]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

PARI

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

Formula

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.