New Sequence from wrong Comment in A083088

[Rainer Rosenthal]

David W. Cantrell wrote:

> With those two corrections, and using your correction to express the 
> formula more nicely in terms of (n - 1/2), we should have
> 
> a(n) = floor( (e - 1)(n - 1/2) + (e - 1/e)/(24(n - 1/2)) )
>
> ... 
> Please do at least a small "sanity check" for my doubly corrected 
> formula. 

Dear David,

I did as you asked me to do. Not only since you asked but also because
I was alarmed why my numerical checks didn't find the typo "e+1/e"
[instead of "e-1/e"] in your first version.

I see more clearly now. The difference is too small due to the n in the
denominator. It was interesting for me to see how your ingeniously
constructed formula could work properly even with that little typo.

I even dared to suppress the second term completely:

          Simple(n) = floor((e-1)*(n-1/2))

Just for fun. And there were only n=1 and n=36 with wrong values. I became
curious and deduced other candidates for failure: n = 9045, 5195512, 5311399545
from the continued fraction expansion of e. As expected, the Simple(n) values
are different from a(n). By the same method but with other candidates, I
checked if there were n such that the first version "e+1/e" produced wrong
values. But I didn't find any. Again, the difference is very small compared
to the difference between (e-1)*(n-1/2) and the nearest integer, which is
critical for the result of the floor function.

> Alas, there is nothing deep about that constant. It just follows from 
> the asymptotics of the two sequences. That's why I said "easily 
> deduced" above.

Thanks, it was not really bright from me to get excited over lim(a(n)-b(n))/n
= lim(a(n)/n) - lim(b(n)/n), sorry.

Thanks for the help with "my" sequence which gets in shape by now and has
a great formula already. I am going to submit the sequence tomorrow and
I would be pleased if somebody gave interesting comments, so that I could
cite him (or her) in the submission.

Best regards,
Rainer Rosenthal
r.rosenthal at web.de




Dear SeqFans:

As noted in http://www.ams.org/ams/prizebooklet-2008.pdf, today Neil
received the first David R. Robbins Prize, which honors "the author or
authors of a
paper reporting on novel research in algebra, combinatorics, or discrete mathe-
matics."  Neil's 2003 paper, in Notices of the AMS, describes OEIS.

My hat is off to NJAS!  I wish I could have been at the San Diego meeting
today.

Tony