# Tagged Questions

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How would I prove $$\sum\limits_{i,j\ge 0} {n-i \choose j} {n-j \choose i}=F_{2n+1}$$ where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of Fibonacci numbers? Thank you very ...
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### Number of zeros in Fibonacci sequences mod $p$

We know that Fibonacci sequences are periodic in mod $m$. For example, for $p\equiv \pm 1 \pmod 5$ and $\pm 2 \pmod 5$ the periods for Fibonacci sequences modulo $p$ divide $p-1$ and $2p+2$ ...
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### How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
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Let $L_n$ be the Lucas numbers, defined by $L_n = F_{n-1} + F_{n+1}$ where f is the Fibonacci numbers. How to prove that $L_{2n+1} = \displaystyle \sum_{k=0}^{\lfloor n + 1/2\rfloor}\frac{2n+1}{2n+1 ... 1answer 95 views ### How to calculate variety of a vector under this constraint? I am currently working through W. Ross Ashby's An Introduction to Cybernetics, and I'm stuck on a problem of calculating variety for a vector. I know the answer (I caved and checked), but I can't ... 3answers 2k views ### Fibonacci sequence divisible by 7? Make and prove a conjecture about when the Fibonacci sequence,$F_n$, is divisible by$7$. I've realized it's when$n$is a multiple of$8$. I just don't know how to go about proving it. 3answers 270 views ### Fibonacci Numbers: Is This Notation Clear? How Can It Be Improved? I am writing up an assignment with includes many identities of Fibonacci numbers. I have made up the following notation (here$f_n$is the number of tilings of an$n$-board by dominoes and squares - a ... 1answer 333 views ### Lucas Numbers and Tilings Show that$f_{n-1} + L_n = 2f_{n}$. So we need to find a$2$to$1$correspondence. Set 1: Tilings an$n$-board. Set 2: Tiling of an$n-1$-board or tiling of an$n$-bracelet. So we need to ... 1answer 1k views ### Combinatorial proof of a Fibonacci identity:$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$Does anyone know a combinatorial proof of the following identity, where$F_n$is the$n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it ... 1answer 347 views ### What is the least upper bound on the number of pairs of disjoint subsets of a binary step-order set S whose sums differ by 1? Define a binary step-order set as a finite set S of positive integers si, i = 0 to n-1, where si ≡ 2i (mod 2i+1) for each i from 0 to n-1. So, i is the power of 2 appearing in the prime factorization ... 1answer 2k views ### Quadratic reciprocity via generalized Fibonacci numbers? This is a pet idea of mine which I thought I'd share. Fix a prime$q$congruent to$1 \bmod 4$and define a sequence$F_n$by$F_0 = 0, F_1 = 1$, and$\displaystyle F_{n+2} = F_{n+1} + \frac{q-1}{4} ...
The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...