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By Flajolet Ph., Sedgewick R.

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For Ð➻➯❴ ò , introduce the class (language) ➦ ❘ of all words Ø such that ✆ ✆ ✤ æ ▲ ✺ ▲ ✺ ✺ the automaton, when started in state ❹ ❘ , terminates in one of the final states after having ✟ read Ø . The following relation holds for any : ó✔é ❛ ❘ ❘ ➦ (33) ïèç ö Ø ❞ ❴✫➾ ➦ ❛③ ê ⑥ ➀ þ ⑩✤ë ➻ Ú ❛ ❘ there ç is the class ❴❝ì formed of the word of length 0 if ❹ ❘ is final and the empty set ( í ) otherwise; the notation ❹ ❘➙î ➾❯û designates the state reached in one step from state ❹ ❘ upon reading letter ➾ .

For the case of a finite è , we predict from Proposition 2 that ① ë ❝ û is always a rational function with poles that are at roots of unity; also the ① ë satisfy a linear ø recurrence related to the structure of è . The solution to the original coin change problem is found to be ✄ ✝ ❝ ➐✳➐ ✞ ð ❑❼ ï ❝ ❝ ❝ ❝ ❸ ❸ ð❤❣ û ð❤❣ û ð❤❣ û ð ❣ ý û ú ú✺ õ õ ø p. 108] ø that ø In the same vein, one ø proves [28, ❑✓ø ❑✦ø ø î✟✄ ö ✐ ✮ î✧ö ✐ û ý ✮ ① ÷ ý✦ù ï✭✬ ① ÷ ý ✸✦ù ï✯✬ ú ø ð ✺ ❑ ❏ ❏ ❏ . Such results are ✱ ✮ ✎ ✰ ✲ ✮ ö ý denotes the integer closest to the real number ú There ✬ typically obtained by the two step process: (i) decompose the rational generating function into simple fractions; (ii) compute the coefficients of each simple fraction and combine them to get the final result [28, p.

That do not ❿ ❣➽ð , a quantity that grows at an exponential rate of þ , with þ ï contain ✂ ✈ ➉ ➊❻✷ ➊ is ú ✸ ðùö û the golden ratio. Thus, all but an exponentially vanishing proportion of the strings of length ø ú î contain the given pattern ➉ ➊❻➊ , a fact that was otherwise to be expected on probabilistic grounds. ) þ ➊❻➊ This example is simple enough that one can also come up with an equivalent regular expression describing ➦ ❼ : an accepting path in the automaton of Figure 8 loops around state 0 with a sequence of , then reads an ➉ , loops around state 1 with a sequence of ➉ ’s ➊ and moves to state 2 upon reading a ; then there should be letters making the automaton ➊ passs through states 1-2-1-2- ❖❀❖▲❖ -1-2 and finally a followed by an arbitrary sequence of ➉ ’s ➊ and ’s at state 3.

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