I don't know how to solve recurrence equation with positive indices like $$a_{n+2}  + 4a_{n+1}+ 4a_n = 7$$ by generating functions. How to solve such kind of problems.
So I had asked a question prior to this one about recurrence relations, but apparently it was a bad one to ask. So I'm trying again to understand how to solve these babies... Here it is:  $$ 3a_{n+1}-4a_n=0, n&gt;=0, a_1=5 $$  What is the general process for solving a relation like this?
I've removed the poll style answers in favour of open discussion.  I realise now this isn't the best way to have a proper discussion about this issue - sorry to anyone who commented/voted.    This is mostly a followup on this question:    As I've answered, the reason this is happening is that the root database (in the production Stack Exchange database server) is still called StackExchange.Programmers - a database name that propagates to SEDE and through it to the data dump.  The question here is what to do about it - the current situation is somewhat broken for users of the data dump.  These are the options we have:   Do nothing Update the root database name (requires downtime - so very undesirable) Update the update scripts so the SEDE database name is the new name, change will flow to the data dump export (breaks any queries using the old name) Update the export to the data dump so only it gets the new name Something else we haven't thought about   I have added all of these options as answers below - please vote on the one you most prefer, we will try to accommodate the top voted answer. The annual student intake of Hogwarts seems about 100. Since in book 7 Hogwarts is declared mandatory for all children, this means that the total magical population of Britain cannot be much more than 10,000.  How do these people make a living?  All jobs mentioned in the books are non-productive occupations (government, stores, sports &amp;c) - not farmers or engineers. There are exceptions - makers of wands/robes/broomsticks - but where do they get the raw materials from?  Who and how "creates goods" in the wizarding world?  Note that if you say "wizards exchange their gold for Muggle money and buy stuff from Muggles (through intermediaries, if necessary)" then you would have to explain where they get their gold: they would either have to mine it, or produce something of value to the Muggles (in which case the separate financial system of galleons/sickles/knuts does not make sense).  PS. A simple issue of maintaining a steam locomotive for Hogwarts express is pretty much insurmountable for 10k people - without Magic, that is. So, I can imagine that the skilled tradesmen can transfigure dirt into raw materials for their trade - except for food, which "is the first of the five Principal Exceptions to Gamp's Law of Elemental Transfiguration". When people are asked to match monchromatic violet light with an additive mix of basic colours, they (paradoxically) mix in red.  In fact, the CIE 1931 color space chromaticity diagram shows this effect begins at about 510nm (greenish-cyan), where people mix in no red.  From that point on, the higher the frequency of the light source, the more red they mix in.    This effect is reflected by the red curve of the , which has an additional bump in the area of blue light.  However, that curve does not match the actual spectral sensitivity of red cones.  So where does this additional perception of red at higher frequencies come from? How to Measure Luck vs Skill in Games? I wanted to upgrade to 14.04.2 point release from 14.04.  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Can I get tourist visa on arrival at Dubai airport? How can I combine multiple rows into a comma-delimited list in Oracle? I have a simple query:  select * from countries   with the following results:  country_name ------------ Albania Andorra Antigua .....   I would like to return the results in one row, so like this:  Albania, Andorra, Antigua, ...   Of course, I can write a PL/SQL function to do the job (I already did in Oracle 10g), but is there a nicer, preferably non-Oracle-specific solution (or may be a built-in function) for this task?  I would generally use it to avoid multiple rows in a sub-query, so if a person has more then one citizenship, I do not want her/him to be a duplicate in the list.  My question is based on the similar question on .  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The likelihood function is: \begin{align} L(\theta) &amp;= f(x_1, \cdots, x_n; \theta) \mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{f(x_i ; \theta)}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{(\theta+1)x_i^{\theta}}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\} \\ &amp;=(\theta+1)^n \left( \prod_{i=1}^n{x_i}\right)^\theta \end{align}  We look at the log likelihood $l(\theta)$ and take the derivative of this since it is easier to deal with and it is allowed because the log is monotonic:  \begin{align} l(\theta) = n\log(\theta+1) + \theta \left( \sum_{i=1}^n{\log(x_i)} \right) \end{align}  $\implies \frac{d}{d\theta} = \frac{n}{\theta+1} + \sum_{i=1}^n{\log(x_i)} = 0$  $\implies \hat{\theta}_{MLE} = -\frac{n}{\sum_{i=1}^n{\log(x_i)}} - 1$  However, this doesn't look right. So our 6-month-old boy still doesn't sleep through the night. He still wakes up at least once and wants to eat.  He takes 3 naps during the day, ranging from 30 minutes to 1.5 (sometimes 2) hours.  We usually put him down for the night at least 2 hours after his latest nap, but we haven't been able to get him down to 2 naps per day because he gets pretty cranky during the evenings. Depending on the duration and times of his naps, his bedtime ranges between 10 and midnight.  We have started feeding him a small amount (2 tablespoons dry) of rice cereal after his morning and evening naps in the hopes that he'll have a fuller stomach through the night, but he still wakes up hungry after sleeping between 3 and 6 hours. I can handle the 6 hours, since that means I get 6 hours of pretty much uninterrupted sleep, but it's usually more like 3-4.  Any suggestions? LINQ's Distinct() on a particular property I am playing with LINQ to learn about it, but I can't figure out how to use Distinct when I do not have a simple list (a simple list of integers is pretty easy to do, this is not the question). 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