Only finitely supported sequences have rational Lambert series among those that are eventually linearly recurrent.
Rivin,Walks on Free Groups and other Stories – twelve years later, arXiv:1106.5947, 2011; published as Growth in free groups (and other stories)–twelve years later,Illinois J
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Linear recurrences and rational Lambert series
Only finitely supported sequences have rational Lambert series among those that are eventually linearly recurrent.