query: I want to find the closed form of $\sum\limits _{i=0}^{\lfloor n/2 \rfloor} {n-i \choose i}$
positive_passages.docid: 18046
positive_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
negative_passages.docid: 40779
negative_passages.text: What is the smallest possible value of $\lfloor (a+b+c)/d\rfloor+\lfloor (a+b+d)/c\rfloor+\lfloor (a+d+c)/b\rfloor+\lfloor (d+b+c)/a\rfloor$?
negative_passages.docid: 110441
negative_passages.text: Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
negative_passages.docid: 92754
negative_passages.text: Closed form for the series: $\sum_{n=1}^{\infty}x^{n^2}$
negative_passages.docid: 126304
negative_passages.text: Determining bounds for the sum $\sum\limits_{n=1}^\infty \frac{1}{2^n - 3^n }$
negative_passages.docid: 138092
negative_passages.text: What is $\lfloor i\rfloor$?
negative_passages.docid: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
negative_passages.docid: 118571
negative_passages.text: Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
negative_passages.docid: 2877
negative_passages.text: Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$
negative_passages.docid: 102486
negative_passages.text: Simplifying $\sum_{i=1}^n{\lfloor \frac{n}{i} \rfloor}$?
negative_passages.docid: 146651
negative_passages.text: Evaluting sum $\sum \limits_{n=0}^\infty\frac{n^k}{n!}$
negative_passages.docid: 45862
negative_passages.text: Nested sum of binomial coefficient in closed form
negative_passages.docid: 90030
negative_passages.text: What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$
negative_passages.docid: 140462
negative_passages.text: Find a closed form for the power series $f_p(x)=\sum\limits_{n=0}^\infty\frac{n^p}{n!}x^n$
negative_passages.docid: 145798
negative_passages.text: Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$
negative_passages.docid: 89698
negative_passages.text: Please help with $\sum\limits_{r=1}^{N}\frac{2r+3}{3^r(r+1)}$
negative_passages.docid: 151012
negative_passages.text: Find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$