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I am becoming more familiar with proving sequences that are given in a problem, though I am not familiar with how to actually define a sequence. More specifically, I am dealing mostly with binary numbers and tiling problems, though again, most of what I have deslt with have been proofs, not trying to define a sequence.

The particular problem that has been frustrating me for the past few days is this:

  • Define a sequence {$s_n$} by recursion such that there are s_n different sequences of 0's and 1’s of length n that do not contain three consecutive 1's.

I know the answer is $s_n = s_{n−1} + s_{n−2} + s_{n−3}$ but I don't understand why. I have tried listing the possible values for n:

  • n = 1 has 2 posibilities (0, 1)
  • n = 2 has 4 posibilities (00, 01, 10, 11)
  • n = 3 has 7 posibilities (000, 001, 010, 011, 100, 101, 110, not including 111)
  • etc.

I know this problems also relates to Fibonacci's sequence in that each number in the sequence builds off of the previous one(s), though at this point, I don't know where to continue. Could someone please explain the steps to get to $s_n = s_{n−1} + s_{n−2} + s_{n−3}$?

AMM
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1 Answers1

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A sequence of $0$'s and $1$'s with no three consecutive $1$'s, of length $n>3$, has a form of $s0$, or $s01$, or $s011$, where $s$ is again a sequence of $0$'s and $1$'s with no three consecutive $1$'s (of length $n-1$, $n-2$ or $n-3$, respectively). And any such $s$ results in a valid sequence; we have one-to-one correspondence.

metamorphy
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