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For a row $n$ of Pascal's triangle, what is the expected value of the sum of every $k$th element, (Starting from $1,2,\cdots,k-1$ places), as compared to $\frac{2^n}{k}$?

I've tried representing the sums of each $k$ in terms of roots of unity, but since there's no clear evaluation of the roots of unity with ease, I haven't really gotten anywhere that way.

For 3, I found a pattern that repeats every 6 rows away, but the pattern seems to diverge as $k$ increases. Is there a clear pattern or a way to represent this in a cleaner formula?

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    Its a numbers, not a random variable, what do you mean? – 2017-01-14
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    Like I mean could the distance from the sum to $\frac{2^n}{k}$ be expressed in terms of $n$ and $k$ I guess is what I'm saying. – 2017-01-14
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    If you want ${n\choose0}+{n\choose3}+{n\choose6}+\cdots+{n\choose3[n/3]}$, it's $(1/3)(2^n+2\cos(n\pi/3))$. See http://oeis.org/A024493 – 2017-01-17
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    Thanks for the resource. Is there a way to extend this to a higher $k$? – 2017-01-17
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    For ${n\choose0}+{n\choose4}+\cdots+{n\choose4[n/4]}$, http://oeis.org/A038503 gives $a(n)=2^{n-1}+2^{(n-2)/2}(\cos(\pi n/4)-\sin(\pi n/4))$ but this looks like $(1/2)2^n$ instead of $(1/4)2^n$ so I would suggest caution. – 2017-01-18
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    See also https://www.math.hmc.edu/~benjamin/papers/EvenlySpacedBinomialsMag.pdf and J. Konvalina, Y.-H. Liu, Arithmetic progression sums of binomial coefficients, Applied Mathematics Letters Volume 10, Issue 4, July 1997, Pages 11–13. – 2017-01-18
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    Also worth a look: https://arxiv.org/pdf/1610.09361v1.pdf and http://math.stackexchange.com/questions/287208/determining-when-a-certain-binomial-sum-vanishes – 2017-01-18
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    Just one more reference: Johnson et al., Pascalian rectangles modulo $m$, Quaestiones Math 14 (1991), no. 4, 383-400, MR1143043. – 2017-01-18
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    So, qt., have you had a look at those links? – 2017-01-19
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    I'm in the process of reading the 2nd to last link, and the other paper did help in confirm the formula I derived through roots of unity. Since there doesn't really seem to be a clear pattern for a general $k$th binomial coefficient, I have a feeling that the question may have to specifically do with the relationship it has with the expected value. – 2017-01-19

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