Re: -- A sequence or real numbers

On Feb 13, 5:30 pm, c...@xxxxxxxxxxxxxxx wrote:
Does anyone know, or at least have a clue on the following question ?

Let (b_n)_n be the sequence defined by :

b_1=1/4 , b_2=1/16 and

b_{n+1}=b_1 b_n+b_2 b_{n-1}+ ... +b_k b_{n-k+1}+ ... +b_1 b_n

Ex: b_3=b_1 b_2+b_2 b_1=1/32
b_4=b_1 b_3+b_2 b_2+b_3 b_1=5/256 etc.

Finally, let

X_n=b_n . n^{3/2}

A computer evaluation hints to the fact that (X_n) is decreasing to a
limit approx equal to 1.141...

Proof (?)

Note: the initial values (i.e. b_1,b_2) seems to be essential. Any
change, and (X_n)_n appears
to behave chaotically. [again, computer says so ]

Any information is warmly welcome.
Thanks
cosv

Define a (formal) series putting

f = sum_{n >= 1} b_n t^n

One can easily see this actually has a positive convergence
radius. Your recurrence relation means that

f^2 = f - 1/4

Solving this tells us that

f = (1 - sqrt(1 - t)) / 2

Therefore you can compute the b_n exactly by
figuring out the derivatives of f:

b_n = f^{(n)}(0) / n!

The derivatives at zero have a simple form,
which will let you find what you want.

-- m

.

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