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How do I show that the following sum is equal to the fibonacci numbers? Atleast numerical evaluation suggests it is $$ \sum_{k=0}^{\lceil n/2\rceil}\binom{n+1-k}{n+1-2k} $$ The image below shows how it moves through pascals triangle, it starts at the right $1$ on line $n+1$, if the upper line is $n=0$. enter image description here

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    $\begingroup$ Use induction, showing that your formula satisfies the Fibonacci recurrence relation $F(n+1) = F(n) + F(n-1)$. $\endgroup$ – Bob Krueger Apr 5 '15 at 19:52
  • $\begingroup$ @Bob1123 right, that was pretty obvious. Oh well $\endgroup$ – Alice Ryhl Apr 5 '15 at 19:54
  • $\begingroup$ It may be easy to show that the relation is true, but it does not make the identity obvious. Take a look at the last picture here. $\endgroup$ – Bob Krueger Apr 5 '15 at 20:03
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It's clear that: $$ \binom{n+1-k}{n+1-2k}=\binom{n+1-k}{k} $$ and there is a solution here

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