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21 problems match “recurrence relations”. Clear

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?
Fifth term of a recursive sequence ●○○○○ 10 ptsIn the sequence $\{a_n\}$ , $a_1=2$ and $a_{n+1}=2a_n-1$ . Find $a_5$ .
Blasting an asteroid into pieces ●○○○○ 10 ptsBruce the Astronaut must destroy a large asteroid. When he blasts a large asteroid, it splits into two medium asteroids. When he blasts a medium asteroid, it splits into three small asteroids. When he blasts a small asteroid, it vanishes. How many blasts does he need to make the asteroid completely vanish?
First four terms of a recurrence ●○○○○ 10 ptsFind the first four terms of the sequence defined by $t_n=2t_{n-1}+4$ with $t_1=8$ .
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 $A(2, 1)$ .
Doubling map sequence term ●●○○○ 20 ptsA sequence $\{a_n\}$ satisfies $a_{n+1}=2a_n$ if $0 \leqslant a_n < \frac{1}{2}$ and $a_{n+1}=2a_n-1$ if $\frac{1}{2} \leqslant a_n < 1$ , for every positive integer $n$ , and $a_1=\frac{6}{7}$ . Find $a_{24}$ . Round to two decimal places.
Seven-letter good words ●●○○○ 20 ptsDefine a good word as a sequence of letters that consists only of the letters $A$ , $B$ , and $C$ (some of these letters may not appear in the sequence), and in which $A$ is never immediately followed by $B$ , $B$ is never immediately followed by $C$ , and $C$ is never immediately followed by $A$ . How many seven-letter good words are there?
Count unival subtrees in a tree ●●○○○ 20 ptsA unival subtree of a binary tree is a subtree in which all nodes have the same value. A single leaf counts as a unival subtree. Consider the following binary tree. The root has value $0$ . The root's left child has value $1$ and is a leaf. The root's right child has value $0$ ; its left child has value $1$ and its right child has value $0$ (a leaf). The node with value $1$ (the left child of the right child) has two children, both with value $1$ (leaves). How many unival subtrees does the tree have?
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 $a_1$ .
Integer trapped between two sequences ●●●○○ 50 ptsLet $\{a_n\}$ and $\{b_n\}$ be sequences with $a_1=0$ and $b_1=2013$ , such that for every positive integer $n$ the triples $(a_n, a_{n+1}, b_n)$ and $(a_{n+1}, b_{n+1}, b_n)$ are arithmetic sequences. Find the unique positive integer $c$ such that $a_n < c < b_n$ for all positive integers $n$ .