[seqfan] Pi Day Question

[Leroy Quet]

[Whoops. I accidently sent this to seqfan at seqfan.com. Resending to correct address now...]

Happy Pi Day, everybody! (At least in the US, where it is still March 14 and where the date is written 3/14.)

A pi question:

Consider the simple continued faction of pi. (The terms of which are sequence A001203.)

The comment at the related sequence A032523 (A032523(n) = the index of the first occurrence of n in A001203) suggests that it is not known for certain that every positive integer occurs in the simple continued fraction of pi.


Can it be said, however, that either: it is known that each positive integer that does occur occurs infinitely often; OR it is known that at least some integers occur finitely often?


I am not even sure, from what little I have read, that if is known that 1 occurs infinitely often in the continued fraction of pi.

Can anyone enlighten me (and us)?

Thanks,
Leroy Quet