Consider sequences $x$ of length $n$ with binary values $x[i] = \pm 1$. Compute the linear autocorrelation coefficients $a[k] = \sum_{i=1}^{n-k} x[i] x[i+k]$. A binary sequence with all nonnegative autocorrelation coefficients has $a[k] \ge 0$ for all $k = 0\cdots n-1$.
Give the number of such binary sequences amongst the $2^n$ possible sequences, where these numbers are dependent on $n$.
With computer evaluations, here are the numbers for $n = 2 \cdots 20$:
2 2 2 6 10 12 14 24 52 74 98 170 326 506 712 1192 2230 3630 5588
It can be seen that the increase rate is not 2, i.e. the numbers do not double (as $2^n$ does) with each rising n. Indeed, the fraction of the $2^n$ possible sequences falls to 0.0508 at n=10 and to 0.0053 at n=20.
Can someone come up with ideas which describe (estimate / limit) this behavior with $n$? Does the fraction fall to 0 as $n \to \infty$? Will the increase rate converge towards some factor less than 2, or even towards 2 eventually?