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For n ≥ 0 let a(n) be the number of quinary strings (only contain digits among 0 . . . 4) of length n and do not contain the string 00. Find a recurrence relation and give initial conditions for the sequence a0 , a1, ...

Completely confused with this question, how do I go about find a recurrence relation for a quinary string?

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Let a string be called valid when it has your property. Let $a_n$ denote what you mentioned. Let $c_n$ denote the number of $n$ digit quinary strings without $00$ with the additional condition that the string does not have a $0$ at the end. Let $b_n$ denote the number of $n$ digit quinary strings without $00$ with the additional condition that the string has a $0$ at the end. It is easy to see that $a_n=b_n+c_n$. Also, one has these relations: $$b_{n+1}=c_n$$ $$c_{n+1}=4a_n=4(b_n+c_n)$$ The first one comes from the fact than any valid string with no $0$ at its end can be extended to a valid string with a $0$ at the end by appending a $0$. For the second one, if we have any valid string then we can append one of $1,2,3,4$ at its end to get a valid string without a $0$ at its end. Using this, we get $c_{n+1}=4(c_n+c_{n-1})$ and $b_{n+1}=4(b_n+b_{n-1})$ and on adding we get $a_{n+1}=4(a_n+a_{n-1})$.

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  • $\begingroup$ Thanks for the quick answer, Manan! Unfortunately, I am still a bit confused with your explanation... $\endgroup$
    – Audrey Kan
    May 11, 2018 at 5:38
  • $\begingroup$ A) In the first part of your answer you say a(n) = b(n) + c(n). So an example of that would be 140230 = 1402 + 30. B) Then, I am also confused with the second relation c(n+1) = 4a(n)... where does the 4 come from? $\endgroup$
    – Audrey Kan
    May 11, 2018 at 5:44
  • $\begingroup$ We need to find $a_n$. First, we break $a_n$ into two parts $b_n$ and $c_n$. $b_n$ counts the number of $n$ digit valid strings with a $0$ at the end, and similarly no zero at the end for $c_n$. We are not breaking up the string itself. We are just counting valid strings with complementary properties and then adding up to get the total number of valid strings. $\endgroup$
    – Manan
    May 11, 2018 at 5:46
  • $\begingroup$ A string contributes to $c_{n+1}$, that is, is of $n+1$ length and is valid without a $0$ at the end if and only if its last digit is not $0$ and the string formed by its first $n$ digits is valid. Now, the last digit has $4$ non zero choices, namely $1,2,3,4$ and the number of valid strings with $n$ digits is just $a_n$. $\endgroup$
    – Manan
    May 11, 2018 at 5:49
  • $\begingroup$ The recurrence is in oeis.org/A086347 . $\endgroup$
    – R. J. Mathar
    Feb 13 at 13:35

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