Based on the comments on OEIS-A002387 (https://oeis.org/A002387):

$a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,...

it is likely, that the sequence $a_{n}$ coincides with $\lfloor e^{n-\gamma} +0.5\rfloor$ , where $\gamma$ is the Euler-Mascheroni constant (remark made by Dean Hickerson).

My Question: Is there a formal proof, that OEIS-A002387 is $\lfloor e^{n-\gamma} +0.5\rfloor$ ?