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I was given the next question: I need to find the recurrence relation, $An$ , for the number of the finite strictly decreasing sequences which consist numbers that are smaller than $n$ or equal to $n$ and also the difference between two successive numbers is at least 2.

For exmaple, if n=3 the sequences are: (3,0) ,(3,1), (2,0) , (0), (1), (2), (3) therefore $A3$ equals to 7

I'm trying to figure out what's wrong with my solution. I suggested that the problem is similar to the problem of finding how many $n$ long binary strings are there such that between each $'1'$ there is at least one $'0'$ but the answer doesn't seem to be right...

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For $n\geq 2$ we have $A_n=A_{n-1}+A_{n-2}+1$. Why?

The number of decreasing sequences that do not contain $n$ is clearly $A_{n-1}$. The number of sequences that contain $n$ is $A_{n-2}+1$, since each one is obtained by taking a decreasing sequence with a subset of the terms $\{n-2,n-1,\dots,1\}$ and adding $n$ at the beginning, the only one that cannot be obtained this way is the sequence that only contains $n$.

Using this we have the first few values are:

$A_0=1,A_1=2,A_2=4,A_3=7,A_5=12,\dots$

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    Thank you so much Jorge! Can you please explain to me what's wrong with my answer? – 2017-01-23
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    Your observation looks good, although you didn't provide a recurrence. You just said that the two problems have the same answer. – 2017-01-23
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Your solution is almost right ... except for the fact that you need at least a $1$ in your binary string (the empty sequence does not qualify, or does it?), and your string is really $n+1$ bits long rather than $n$ bits long (because you count down from $n$ to $0$, inclusive, and that's $n+1$ numbers)! Can you write the recurrence relationship for your case?

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    The empty sequence does qualify, $A0$ = 1 Oh..the problem was with the string's length..ahh..the small things that can ruin a proof. Thank you for pointing it out!! – 2017-01-23
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    Note that the sequence that contains the number "0" is not the empty sequence (it contains one number, the number "0")! The empty sequence is that which contains no numbers at all, not even "0". It's still a sequence, but I'm under the impression that you are not supposed to count it (and your A(3)=7 would confirm you are not supposed to count it), even though from the strict statement of your problem it should qualify :) – 2017-01-23
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This is sequence A000071 in the Online Encyclopedia of Integer Sequences. See: https://oeis.org/A000071 and the references therein for everything you could ever want to know about it.