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I'm stuck on this Math problem :


How many solutions does the equation

$x_{1} + x_{2} + 3x_{3} + x_{4} = k$

have, where $k$ and the $x_{i}$ are non-negative integers such that $x_{1} \geq 1$, $x_{2} \leq 2$, $x_{3} \leq 1$ and $x_{4}$ is a multiple of 6.


I tried to write the possible cases for $x_{2}$ and $x_{3}$ since they are bounded.

Case $x_{2} = 2$ :

  1. Case $x_{3} = 1$ : $x_{1} + 5 + x_{4} = k$
  2. Case $x_{3} = 0$ : $x_{1} + 2 + x_{4} = k$

Case $x_{2} = 1$ :

  1. Case $x_{3} = 1$ : $x_{1} + 4 + x_{4} = k$
  2. Case $x_{3} = 0$ : $x_{1} + 1 + x_{4} = k$

Case $x_{2} = 0$ :

  1. Case $x_{3} = 1$ : $x_{1} + 3 + x_{4} = k$
  2. Case $x_{3} = 0$ : $x_{1} + x_{4} = k$

But now, I'm stuck. Should I try to resolve all this equations ? Am I in the right direction ? Any help would be grealty appreciated.

EDIT :

I found that the number of solutions will be given by the coefficient $a_{k}$ of $x_{k}$, i.e :

$(x^0+x^1+x^2)(x^0+x^1)(\displaystyle\sum\limits_{k=1}^\infty x^k)(\displaystyle\sum\limits_{k=0}^\infty x^{6k})$
$=(x^0+x^1+x^2)(x^0+x^1)(\frac{1}{(1-x^3)(1+x^3)})(\frac{x}{1-x})$
$=\frac{x+2x^2+2x^3+x^4}{(x^3-1)(x^3+1)(x-1)}$

Now I don't know how to find the $a_{k}$ coefficient.

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If you had a particular $k$, your approach is reasonable. However you're trying to solve this for all $k$ at once, i.e. get a function of $k$. There are other methods that should be used, ones that were probably taught right before this problem was posed. –  vadim123 May 22 '13 at 13:10
1  
To see a solved problem of this type, see math.stackexchange.com/questions/266029/… –  vadim123 May 22 '13 at 13:15
    
@vadim123 Thanks, I will take a look –  user2336315 May 22 '13 at 13:18
    
I think the $1+x$ term is wrong, on account of the coefficient of $x_3$ in the problem statement. –  Gerry Myerson May 22 '13 at 13:36
    
I don't understand why you have $(1+x^3)$ instead of $(1+x)$. And what should I do after this ? –  user2336315 May 22 '13 at 13:41

1 Answer 1

up vote 3 down vote accepted

This will give you a start: it's the coefficient of $x^k$ in $$(x+x^2+x^3+\dots)(1+x+x^2)(1+x^3)(1+x^6+x^{12}+\dots)$$

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Check my edit please –  user2336315 May 22 '13 at 13:34

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