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I have started to encounter with recurrence relations quite often now and am unable to find a good source which explains all the varieties of recurrence relations(say homogeneous and so on) and the methods to solve them. So far I only know one method which works out for me and that is manually generate a sequence and find a pattern and using the characteristic equation as well. Sometimes when there's an initial condition given I get confused, here's one example to be specific;

$$a_{n+1}=2a_n+\sqrt{a_{n}^2+a_{n-1}^2}$$

for $n>0$ and $a_1=1$,

Using my rudimentary knowledge, I just started off with plugging $n=1$, to get; $$a_{2}=2a_1+\sqrt{a_{1}^2+a_{0}^2}$$

Then I was left confused to what $a_0$ is and how to deal with it.

I am interested in book(s) which clear such "silly"/"rookie" concepts and also explain all the varieties of recurrence relations(say homogeneous and so on) and the methods to solve them, alongside I am also interested in book(s) which come with a good(tough) problem set to work on after finishing with the theory.

NadiKeUssPar
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    The thing to search for is "difference equations." I've used the book by Kelley and Peterson: https://www.amazon.com/Difference-Equations-Introduction-Walter-Kelley/dp/012403330X – B. Goddard Aug 02 '23 at 14:00
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    Another great book which handles recurrence relations extensively is Concrete Mathematics by Graham, Knuth, and Patashnik: https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf – Matt Werenski Aug 02 '23 at 14:03
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    If you have a recurrence relation that relates $a_n$ to its two previous values, then you need two initial values of the sequence to fully determine it. The Fibonacci sequence is defined by "$F_n=F_{n-1} + F_{n-2}$ and $F_0 = 0, F_1 = 1$", not just "$F_1 = 1$". The reason you got confused by $a_0$ is that you were trying to calculate a sequence that was not fully defined. Decide fully how the sequence should begin, and then you will be able to calculate it. – Paul Sinclair Aug 03 '23 at 16:49
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    @ayan: You might find this answer helpful. – Markus Scheuer Aug 05 '23 at 16:48

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