Investigating constant C=0.1688... in A081881 comment

[Rainer Rosenthal]

Gerald McGarvey wrote:
> An observation about A081881: for n=2 to 17 either
>    a(n) = floor(exp(1)*a(n-1)) - 1 
> or  
>    a(n) = floor(exp(1)*a(n-1))   (for n=2,4,16)

Here are more terms for n = 2, 4, 16, ...

2, 4, 16, 24, 26, 38, 46, 50, 52, 58, 62, 63, 65, 70, 73, 78, 79,
[ see my other post today ]
9894, 9897, 9902, 9918, 9925, 9926, 9931, 9935, 9954, 9958,
9961, 9964, 9966, 9973, 9981

(based on my list of the first 10000 entries of A081881)

I should have added: ... *and* based on the assumption that
David Cantrell's heuristic formula for A136617 is correct.

The sequence A081881 seems to make a good test case for
David's heuristics since A136617 is used for quite large exponents.
It would be great if by different reasoning there could be
confirmation for statements like

     floor(A081881(9980)) = A081881(9981)

where the values are quite large: ln(A081881(n)) = n-2+0.221293.

Cheers,
Rainer Rosenthal
r.rosenthal at web.de