3
votes
1answer
35 views

Can you prove this recursive multiple $n$-sided dice throwing statement?

Let $W_{s,r,n}$ be the total number of ways that the sum $s$ can be displayed after throwing $r$ number of $n$-sided dice. Define $$W_{s,0,n} = \begin{cases} 1, & \text{if s = 0} \\ 0, & ...
0
votes
0answers
30 views

Closed form for a special recursion?

Does the recurrence relation $$ a(n+1) = a(n)^2 + 1,\quad a(1)=1, $$ have a closed form solution? I have tried hard to find it, but failed. Any ideas ? I am particular interested in prime ...
4
votes
1answer
69 views

Number of times $g(p_1)$ occurs in $\sum_{d\mid n}g(d)$

$$ g(n)=\begin{cases} 1 & \text{if }n=1 \\[10pt] \sum_{d\mid n,\ d\ne n} g(d) & \text{else} \end{cases} $$ How can I calculate $g(n)$ efficiently? I was trying to collect all the $g(p)$ ...
3
votes
2answers
321 views

Solving a mathematical recursion to find explicit function

I came across the recursive sequence $$ \begin{align} a_{n+1}&=(r-2)a_n+(r-1)b_n\;,\\ b_{n+1}&=a_n\;, \end{align} $$ and the explicit formula $$ a_n=(-1)^n(r-1)+(r-1)^n\;. $$ I saw that ...
4
votes
3answers
1k views

How can I explain this integer partitions function recursion?

How to explain how this algorithm works? I need to write an article about this but I can't explain why this recursion works fine. It defines the number of partitions of a given integer ...