# [seqfan] Re: paperfolding and continued fraction expansion

Olivier Gerard olivier.gerard at gmail.com
Fri May 7 09:49:20 CEST 2010

```Thanks for your first message on the list, Dimitri.

The question you ask is typical of one of the uses of seqfan: discussing
relations between sequences and ensure the best possible editing of the
Encyclopedia,
with links to the existing litterature.

The topics you list in your first paragraph are all parts of the purpose of
seqfan.

Many seqfans are academic, with a strong majority of mathematicians, but
more than
one third of seqfans are not "professionals" in the classical sense of
researcher
or teacher in an academic institution such as a college or university.

I cannot be more precise because I do not ask this kind of things before
subscription:
genuine interest in integer sequences and basic netiquette are the only
requirements.

We have chemists, physicists, computer scientists, students, musicians,
hobbyists with various backgrounds and ages without disjonction between
these categories.

Welcome on seqfan,

Olivier GERARD

On Thu, May 6, 2010 at 17:03, Dimitri Hendriks <diem at cs.vu.nl> wrote:

> Hi,
>
> I am new to this list, and wonder what it is you discuss here. Is it mainly
> classification etc. in a wider sense? And, are all "seqfans" mathematicians
> (I am not)?
>
> (and so A007400):
> I experimentally found (no proof yet) that the sequence of run lengths of
> the regular paperfolding sequence (A014577) is related to A007400, the
> continued fraction expansion of the sum of the series 1/2,1/4,1/16,1/256,...
> . Namely it is sequence A088431, which is the tail^2 of A007400 with the
> values halved.
>
> A "run" is a maximal subword of consecutive identical digits. Thus, writing
> a for A014577 and b for A088431, it appears that we have
>
>  b(n) = length of n-th run of a ,
>
> or as a haskell program:
>
> b = rls a
>
> where rls is a function mapping bitstreams to streams of nats defined by:
>
> rls xs = rls_0 xs 0
> where
> rls_0 (0 : xs) n = rls_0 xs (n+1)
> rls_0 (1 : xs) n = n : rls_1 xs 1
> rls_1 (0 : xs) n = n : rls_0 xs 1
> rls_1 (1 : xs) n = rls_1 xs (n+1)
>
> Is this relationship between the paperfolding sequence and the continued
> fraction expansion corresponding to the sum of the series 1/2^(2^n) a know
> relationship? Is there a proof?
>
> Thanks,
> Dimitri Hendriks
>
>
>
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