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Suppose $X_1, X_2$ and $X_3$ are all non-negative integers. So for this linear integer equation:

$$X_1 - 2X_2 + X_3 = 10$$

Please note that the coefficient for $X_2$ is negative (i.e. $-2$).

What is the generating function for it?

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    It helps us to know what have you tried. You're apt to get more help if you explain what you know, what you've tried, and/or where you are stuck. Otherwise some people may think you're asking us to do your (home)-work for you.2012-12-10
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    No this is really not related to any HW. My reading on the linear Diophantine question in this form seeking for non-negative solutions only has the non-negative coefficients, and there is no document/materials treating the negative coefficients. http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf's book doesn't study my case either.2012-12-10
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    Even if it were homework, we field homework questions as well. Suggestion: why don't you include the text from your comment (immediately above) directly in your post, so we all can better understand what you're looking for, and where the question originates.2012-12-10
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    Your question only talks about $3$ terms $X_1$, $X_2$, and $X_3$. In my answer, I've extrapolated this to mean any three consecutive terms in a sequence, but it is not clear from your question that this is the proper thing to do. Would you please clarify the question?2012-12-11

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