Investigating constant C=0.1688... in A081881 comment

[David W. Cantrell]

Rainer Rosenthal wrote:
> David W. Cantrell wrote:
>
>>>>
>>>>             a(n) = 1 + floor(C*exp(n))                 (1)
>>>>
>>>>    where
>>>>
>>>>      C = 0.1688563566671442037316797755009010341...    (2)
>
>> In <http://www.research.att.com/~njas/sequences/A081881>,
>> Benoit Cloitre had already said that "a(n) is asymptotic to
>> C*exp(n)
>> where C=0.1688...", which I suppose that has been proven. My simple

I should, rather, have said _simplistic_.

>> numerical investigation seemed to indicate that
>>
>>               0 < a(n) - C*exp(n) < 1.                   (3)
>>
>> I don't know if that's always true, but if it is, then of course
>> (1)
>> holds for all n.
>
> Based on your formula for A136617 we can do more such simple
> numerical
> investigations in a reasonable amount of time. I did so and I found
> that (3) doesn't hold for n = 165. I will demonstrate this now.

Many thanks for your investigations! I confirm your observation.

Best regards,
David