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I was asked to find a recurrence relation for the number of n-digit sequences above $\{1,2,3,4,5,6,7,8\}$ such that two successive numbers can't be 'evenly successive' - for example, $4$ can't come after $2$

I'd really like some clues on this question, thank you very much!

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    "$4$ can't come after $2$, but $8$ can come after $2$" - that doesn't really explain a lot. How exactly are we supposed use this ill-phrased definition in order to conclude what digit can or cannot come after every digit??? What does "evenly successive" even mean? And is that symmetric??? i.e., is $f(x,y)=f(y,x)$? – 2017-01-24
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    The question is fixed – 2017-01-24

1 Answers 1

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Let $a_n$ denote the number of sequences ending with an even digit.

Let $b_n$ denote the number of sequences ending with an odd digit.

Then:

  • $a_1=4$
  • $b_1=4$
  • $a_n=4b_{n-1}$
  • $b_n=4a_{n-1}+4b_{n-1}$

Let $c_n$ denote the number of sequences, then $c_n=a_n+b_n=4a_{n-1}+8b_{n-1}$.

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    Hey barak! I don't know if I was clear enough haha..but did you take into account that, for exaple, 8 can come after 4 but can't come after 6? – 2017-01-24
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    @Lola: No. You said that even digits cannot be successive, so $8$ cannot come after neither $4$ nor $6$. – 2017-01-24
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    I'm sorry! English isn't my first language. What I meant was, that numbers that are successive in terms of them belonging to $Neven$ I fixed the example that I gave above – 2017-01-24
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    @Lola: So after $1$ we allow every digit, after $2$ we allow $2,6,4,8$, after $3$ we allow $6$ and after $4$ we allow $8$? And after $5,6,7$ and $8$ we allow only the same digit? – 2017-01-24
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    @Lola: This contradicts your first comment, so you need to make up your mind! – 2017-01-24
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    @Lola: Nevermind that. You need to explain what you mean by "evenly successive". – 2017-01-24
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    @Lola: I honestly don't understand why you've accepted my answer if it doesn't refer to the definition in your question. Please rephrase this definition so that I can fix my answer to address your question and not something else. – 2017-01-24
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    I misread the question and your answer fits the one i was asked. i'm changing the question now. Thank you again – 2017-01-24