# [seqfan] paperfolding and continued fraction expansion

Dimitri Hendriks diem at cs.vu.nl
Thu May 6 17:03:30 CEST 2010

```Hi,

I am new to this list, and wonder what it is you discuss here. Is it mainly about the encyclopedia, or is it about sequences, their properties, classification etc. in a wider sense? And, are all "seqfans" mathematicians (I am not)?

Then about a possible contribution linking sequences A014577 and A088431 (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 ,

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

```