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 A134041 a(n) = number of binary partitions of the Fibonacci number F(n). 1

%I #17 Nov 19 2020 11:56:30

%S 1,2,2,4,6,14,36,114,450,2268,14442,118686,1264678,17519842,318273566,

%T 7607402556,240151303078,10055927801538,559859566727028,

%U 41582482495661986,4129785050606801246,549628445573614296188,98256218721544814784486,23631541930531250077261282

%N a(n) = number of binary partitions of the Fibonacci number F(n).

%H Alois P. Heinz, <a href="/A134041/b134041.txt">Table of n, a(n) for n = 0..125</a>

%F a(n) = A000123( A000045(n) ) for n>=0.

%p 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: f:= proc(n) local t; t:= ilog2(2*n+1); g(n/2^(t-1), t) end: a:= n-> f(combinat[fibonacci](n)): seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 26 2011

%t g[b_, n_] := g[b, n] = If[b < 0, 0, If[b == 0 || n == 0, 1, If[b >= n, Sum[g[b - t, n] Binomial[n + 1, t] (-1)^(t + 1), {t, 1, n + 1}], g[b - 1, n] + g[2b, n - 1]]]];

%t f[n_] := With[{t = Floor@Log[2, 2n + 1]}, g[n/2^(t - 1), t]];

%t a[n_] := f[Fibonacci[n]];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)

%Y Cf. A000123, A000045.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 02 2007

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Last modified July 12 08:52 EDT 2024. Contains 374239 sequences. (Running on oeis4.)