A000123 - asymptotics of binary partitions

[A.N.W.Hone]

Hi seqfans,

I was in Barcelona last week giving a talk in the CRM, and some people
there told me that they had found the recurrence

a_{2m}=a_{2m-1}+a_m

a_{2m+1}=a_{2m}+a_m

which arose in a problem about orbits of ODEs.

With a_0 = 1  this recurrence generates the sequence

1,2,4,6,10,14,20,...

and upon my return I was pleased to find this sequence as A00123 in the OEIS,
the number of binary partitions (partitions of 2n into powers of 2).

It was of interest to know the asymptotics of this sequence,
and I see lots of references on OEIS; apparently Froberg did
to the most thorough job, but Mahler got the leading order
asymptotics, which is

log a_n ~ K (log n)^2  with K = 1/(2 log 2) \approx 0.721

However, there must be large subdominant terms in the asymptotic
expansion (I haven't yet got hold of Froberg's work), because
having found the first 10^6 terms I notice that

r_n := log a_n / (log n)^2

is still way off the above value of K. For instance

r_n \approx 0.537, 0.544, 0.553 and 0.563

for n around 10^3, 10^4, 10^5 and 10^6.

So the limit must be reached very slowly.

Also I notice that Benoit Cloitre has made the conjecture
that

lim_{n\to\infty} (log n) a_{2n} / (n a_n) = c

exists and equals 1.63...

Now at leading order, the existence of
this limit is compatible with the above value
of K (to see this, take logarithms to see that
log a_{2n} - log a_n ~ log n, so
K (log n + log 2)^2 - K (log n)^2 ~ log n implies
 2 K log 2 =1).

However, from another rough asymptotic calculation (using the
recurrence relation above) I find that the
value of c should be

c=K^{-1}=2 log 2 \approx 1.386

which does not agree with Cloitre's estimate.

For 2n=10^2, 10^3, 10^4, 10^5, 10^6

I calculated s_n = (log n) a_{2n} / (n a_n) and got the approx values

1.637, 1.661, 1.645, 1.623, 1.604 respectively.

My feeling is that given the slow convergence of r_n to K,
the converge of s_n is likely  to be at least as slow, so the value of
1.63 for c cannot be trusted.

Does anybody else have any numerical or theoretical experience
with this old problem, before I go and explore the literature?

Andy Hone