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Let's consider the family of sequences of coefficients in the expansion $$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$

Remark. Evidently, the RHS is a finite sum.

Here is a numerical coincidence for which I ask a direct connection as to why?

QUESTION. It is possible to offer say a recurrence for both sides as to why it is so but is there a more direct connection between the two sides? $$\prod_{k=1}^n\left(4+2\cos\left(\frac{2\pi k}{2n+1}\right)\right) =\sum_{k\geq0}a_n(k)^2.$$

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  • $\begingroup$ I guess the first product should start at $i=0$? $\endgroup$
    – Hjalmar Rosengren
    Commented Oct 25 at 3:53
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    $\begingroup$ oeis.org/A001835 $\endgroup$
    – Nemo
    Commented Oct 25 at 8:27
  • $\begingroup$ @HjalmarRosengren: Thank you, edited now. $\endgroup$
    – T. Amdeberhan
    Commented Oct 25 at 12:37
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    $\begingroup$ Nemo's comment, less tersely stated, is that the trigonometric product is the number of domino tilings of a $3\times 2n$ rectangle. $\endgroup$
    – Timothy Chow
    Commented Oct 26 at 12:54

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