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Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, what are the best known upper and lower bounds?

 Note: the walk can repeat vertices.

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Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, what are there the best known upper and lower bounds?

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# Number of closed walks on an $n$-cube

Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, are there known upper and lower bounds?