Hi.

Could you help me solve these equations? I know how to solve recurrence equations with only [tex]a_k[\tex] and scalars in, and sometimes I can also guess the formula and then prove it by induction. But I have no idea how to go about solving these:

1) $\displaystyle a_{n}= \frac{ a_{n-1} ^{3} }{ a_{n-2} ^{2} }$ $\displaystyle |||a_1=1, a_2=2$

2)$\displaystyle a_n=2 \frac{n-1}{n}a_{n-1}+ \frac{1}{n}||| a_1=1$

3)$\displaystyle a_n=5a_{n-1}-6a_{n-2}+4n-3$ $\displaystyle |||a_1=1, a_2=2$

4)$\displaystyle a_n=5a_{n-1}-6a_{n-2}+3^n$$\displaystyle |||a_1=1, a_2=2$

5)$\displaystyle a_n=5a_{n-1}-6a_{n-2}+n3^n$$\displaystyle |||a_1=1, a_2=2$

6)$\displaystyle a_n=5a_{n-1}-6a_{n-2}+n^2 2^n$$\displaystyle |||a_1=1, a_2=2$

7)$\displaystyle a_n=5a_{n-1}-6a_{n-2}+(n^2+1)3^n$ $\displaystyle |||a_1=1, a_2=2$

And two problems:

1)Find the number of strings length n consisting of numbers 0,1,2 where the following bits differ at most 1. What I mean is that such strins should look like that: 00000000111111222222222111111100000001111110000000 011111000001111122222111

I've tried to find a recurrence pattern(if we begin with 0, then after the 0s we can only put 1s, but after 1s there can be 0s or 2s, but then after 0s and 2s there can be only 1s and there we go again) but I have no idea how to write such a recurrence since there can be any number of 1s,2s 0s used.

2) similar, only this time we form bit strings using numbers 0, 1, 2, 3 and we need to find the number of strings in which 0 and 1 are not next to each other.

Please help.