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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.

%I A132880 #26 Sep 16 2024 12:29:18
%S A132880 1,1,2,4,14,60,450,4964,95982,3037948,170005730,16522010532,
%T A132880 2882717916878,902450057292988,514768747418386946,
%U A132880 537142988843543963620,1033976171696917695108270,3688322935382700002945333884,24514290054855626308893309022498
%N 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.
%H A132880 Alois P. Heinz, Max Alekseyev and Dean Hickerson, <a href="/A132880/b132880.txt">Table of n, a(n) for n = 0..50</a> (terms 0..26 by Max Alekseyev and Dean Hickerson)
%F A132880 a(n) = A000123( (2^(n+1) + (-1)^n - 3)/6 ) for n>=0.
%F A132880 a(n) = a(n-1) + A000123( A000975(n-1) - 1) for n>0.
%e A132880 Let b(n) = A000123(n) = number of partitions of 2n into powers of 2, then the initial terms of this sequence begin:
%e A132880 b(0), b(0), b(1), b(2), b(5), b(10), b(21), b(42), b(85), b(170),...
%e A132880 OSCILLATING PARITY TREE.
%e A132880 a(n) = number of nodes in generation n of the following tree.
%e A132880 Start at generation 0 with a single root node labeled [1].
%e A132880 From then on, each parent node [k] is attached k child nodes with labels having opposite parity to k within the range {1..2k}.
%e A132880 The initial generations 0..5 of the tree begin as follows; the path from the root node is given, followed by child nodes in [].
%e A132880 GEN.0: [1];
%e A132880 GEN.1: 1->[2];
%e A132880 GEN.2: 1-2->[1,3];
%e A132880 GEN.3:
%e A132880 1-2-1->[2]
%e A132880 1-2-3->[2,4,6]
%e A132880 GEN.4:
%e A132880 1-2-1-2->[1,3]
%e A132880 1-2-3-2->[1,3]
%e A132880 1-2-3-4->[1,3,5,7]
%e A132880 1-2-3-6->[1,3,5,7,9,11];
%e A132880 GEN.5:
%e A132880 1-2-1-2-1->[2]
%e A132880 1-2-1-2-3->[2,4,6]
%e A132880 1-2-3-2-1->[2]
%e A132880 1-2-3-2-3->[2,4,6]
%e A132880 1-2-3-4-1->[2]
%e A132880 1-2-3-4-3->[2,4,6]
%e A132880 1-2-3-4-5->[2,4,6,8,10]
%e A132880 1-2-3-4-7->[2,4,6,8,10,12,14]
%e A132880 1-2-3-6-1->[2]
%e A132880 1-2-3-6-3->[2,4,6]
%e A132880 1-2-3-6-5->[2,4,6,8,10]
%e A132880 1-2-3-6-7->[2,4,6,8,10,12,14]
%e A132880 1-2-3-6-9->[2,4,6,8,10,12,14,16,18]
%e A132880 1-2-3-6-11->[2,4,6,8,10,12,14,16,18,20,22] .
%e A132880 Note: largest node label in generation n is A000975(n) + 1.
%p A132880 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
%t A132880 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_ *)
%o A132880 (PARI) {A000123(n) = if(n<1, n==0, A000123(n\2) + A000123(n-1))}
%o A132880 {a(n) = A000123( (2^(n+1) + (-1)^n - 3)/6 )}
%Y A132880 Cf. A000123, A000975.
%K A132880 nonn
%O A132880 0,3
%A A132880 _Paul D. Hanna_, Sep 11 2007