# Tagged Questions

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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### Is the Fibonacci constant $0.11235813213455…$ a normal number?

Recall that a normal decimal number is an irrational number $\alpha \in \mathbb{R}$ such that each digit 0-9 appears with average frequency tending toward $\frac{1}{10}$, each pair of digits 00-99 ...
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### Sum of nth powers of Fibonacci numbers

Is a closed form for $$\sum_{i=1}^n{F_i^k}$$ (where $F_i$ is the $i^{th}$ Fibonacci number and $k$ is constant) known?
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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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### Prove the n-th Fibonacci number is less than $2^n$ for all n greater than zero using strong induction

I need to prove the n-th Fibonacci number is less than $2^n$ for all $n \geq 0$ using strong induction. I have been exposed to the idea that strong induction differs from weak induction in that the ...
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### Limit of fibonacci sequence

Let $f_n$ be the $n$th Fibonacci number. Find constants $a$ and $b$ such that $$\lim_{n\to\infty} \frac{f_n}{a\cdot b^n} = 1$$ I'm somewhat confused on how to approach this problem. I know the ...
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### Linear combination of sequence related to Fibonacci

Let $a= \frac{1+\sqrt{5}}{2}$ and $b= \frac{1-\sqrt{5}}{2}$. Prove that for all $c,d \in \mathbb{N}$ that $g_n = ca^n + db^n$ satisfies $g_{n+2} = g_{n+1} + g_{n}$ for all $n\in \mathbb{N}$. I'm ...
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### Find all positive integers $m>1$ such that any sum of $m$ consecutive Fibonacci numbers is divisible by $m$.

Let $\{u_n\} -$ Fibonacci sequence: $u_1=u_2=1, u_{n+1}=u_n+u_{n-1}, n\ge2$. Find all positive integers $m>1$ such that any sum of $m$ consecutive Fibonacci numbers is divisible by $m$. My work ...
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### Lucas recurrence relation

Lucas recurrence relation is defined as: $V_n = PV_{n-1} – QV_{n-2}$; for $V_0 = 2; V_1 = P$ Here $P$ is positive integer and $Q = {-1, 1}$ or may be $+1$ or $-1$ A Fibonacci Pseudo prime with ...
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### Dominos ($2 \times 1$ on $2 \times n$ and on $3 \times 2n$)

How many ways are there to tile dominos (with size $2 × 1$) on a grid of $2 × n$? How about on a grid of $3 × 2n$?
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### Greatest number in fibonacci sequence with property: sum of digits=index in fibonacci sequence

I came across very interesting sequence based on fibonacci sequence. From fibonacci numbers we choose only elements with digit sum=index in fibonacci sequence. It is very interesting that we most ...
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### derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the ...
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### Relationship between decimal length and Fibonacci number

There are 6 single digit Fibonacci numbers. For all other number of digits in the decimal system, there are either 4 or 5 Fibonacci numbers. For example, between 10000 and 99999 there are 5: 10946 ...