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Computing terms of a recurrence

Solve the first problem 16 problems

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1. Fifth term of Alicia's sequence ●○○○○ 10 ptsAlicia starts a sequence with $m=3$. Her algorithm is: 1. Write down the number $m$ as the first term. 2. If $m$ is even, set $n=\frac{1}{2}m$; if $m$ is odd, set $n=m+1$. 3. Write down the number $m+n+1$ as the next term. 4. Set $m$ equal to the value of the term just written in Step 3. 5. Repeat Steps 2, 3 and 4 until she has five terms, then stop. What is the fifth term of her sequence?
2. Fifth term of a recursive sequence ●○○○○ 10 ptsIn the sequence $\{a_n\}$, $a_1=2$ and $a_{n+1}=2a_n-1$. Find $a_5$.
3. First four terms of a recurrence ●○○○○ 10 ptsFind the first four terms of the sequence defined by $t_n=2t_{n-1}+4$ with $t_1=8$.
4. Evaluating the Ackermann function ●●○○○ 20 ptsDefine a function $A(m, n)$ by $$A(m,n) = \begin{cases} n+1 & \text{if } m = 0, \\ A(m-1, 1) & \text{if } m > 0 \text{ and } n = 0, \\ A(m-1, A(m, n-1)) & \text{if } m > 0 \text{ and } n > 0. \end{cases}$$ Compute $A(2, 1)$.
5. First term of a recurrence with a constant subtraction ●●○○○ 20 ptsA sequence $a_1, a_2, a_3, \ldots$ satisfies $a_n = 3a_{n-1} - x$ for every $n \ge 2$, where $x$ is an integer that is the same for all terms. Given that $a_2 = 7$ and $a_4 = 43$, find $a_1$.
6. Third term of a power-plus-constant sequence ●●○○○ 20 ptsLet $\{a_n\}$ be the sequence given by $a_n = a^n + m$ for positive integers $n$, where $a < 0$ and $m$ are constants. Given $a_1 = 2$ and $a_2 = 4$, find $a_3$.
7. Sequence with a quadratic shift relation ●●●○○ 50 ptsLet the sequence $a_0, a_1, a_2, \ldots$ satisfy $a_0=a_1=11$ and $a_{m+n}=\frac{1}{2}\left(a_{2m}+a_{2n}\right)-(m-n)^2$ for all integers $m, n \geqslant 0$. Find $a_{45}$.
8. Initial terms of a recursive sequence ●●●○○ 50 ptsLet the sequence of positive integers $\{a_n\}$ satisfy $a_{n+3}=a_{n+2}\left(a_{n+1}+2a_n\right)$ for $n=1,2,\ldots$, and $a_6=2288$. Find $a_1$, $a_2$, $a_3$.
9. Possible second terms of a sequence ●●●●○ 100 ptsA sequence $a_1, a_2, \ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible?

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