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counting number of circulant subsequences

Let $k\ge1$ and $m\ge1$ be given integers. For any $x=(x_1,\ldots,x_k)\in\{\pm 1\}^k$, define $f(x)=\#\{1\le j\le k: x_j=x_{j+1}=\cdots=x_{j+m-1}\}$. Question: given $0\le l\le k$, for how many $x\in …
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