Answer by Yuval Filmus for Number of words in the regular language $(00)^*$
Continuing Artem's answer, here is a proof of the general representation. As Artem shows, there is an integer matrix $A$ and two vectors $x,y$ such that$$ s_L(n) = x^T A^n y. $$(The vector $x$ is the...
View ArticleAnswer by Artem Kaznatcheev for Number of words in the regular language $(00)^*$
@Patrick87 gives a great answer for your specific case, I thought I would give a tip of how to find $s_L(n)$ in the more general case of any language $L$ that can be represented by an irreducible DFA...
View ArticleAnswer by Patrick87 for Number of words in the regular language $(00)^*$
For your language, can you take $p_0(x) = 1/2$, $\lambda_0 = 1$, $p_1(x) = 1/2$, $\lambda_1 = -1$, and $p_i(x) = \lambda_i = 0$ for $i > 1$? The Wikipedia article doesn't say anything about the...
View ArticleNumber of words in the regular language $(00)^*$
According to Wikipedia, for any regular language $L$ there exist constants $\lambda_1,\ldots,\lambda_k$ and polynomials $p_1(x),\ldots,p_k(x)$ such that for every $n$ the number $s_L(n)$ of words of...
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