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Update: Initially the question was posted for $a = 1$. Now it has been generalized for any real $a > 0$

What is known about the distribution of the sum of the binomial coefficients over multiple of squares? My experimental data seems to suggest that for a given positive real $a > 0$ $$ s_{n,a} = \sum_{1\leq \lfloor ak^2 \rfloor\leq n}{n\choose \lfloor ak^2 \rfloor}= {n\choose \lfloor 1^2 a \rfloor} + {n\choose \lfloor 2^2 a \rfloor} + \cdots + {n\choose \lfloor r^2 a \rfloor} \approx \frac{2^n}{\sqrt{2an}} $$

Clearly the sum will be dominated by the term closest to the central binomial coefficient which in this case is the square nearest to $n/2$. What I found interesting is the shape of the histogram of the distribution of the ratios of the actual sum to its asymptotic estimate i.e. $\dfrac{s_n \sqrt{2an}}{2^n}$ are similar for all $a$ and look like an acr-sine distribution as mentioned in the comments.

Histogram of distribution for $a = 1$

Question 1: Why does it have an arc-sine like distribution?

Question 2 Where does the spikes occur? E.g. for $a = 1$, the spikes occur roughly at $1 \pm 1/6$.

Related question: What is the sum of the binomial coefficients $n \choose p$ over prime numbers?