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I want represent the fraction $$\frac{a+b+a b c+a b d}{1+a c+b c+a d+b d+a b c d}\qquad\qquad\qquad (1)$$ as a continued fraction. Here $a,b,c,d$ free variables.

I could only get $$\frac{a+b+a b c}{1+b c+a d+b d+a b c d}=\frac{1}{d+\frac{1}{a+\frac{1}{c+\frac{1}{b}}}}$$ However, I cannot repersent the original expression (1).

Vertum
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  • For the first expression, there's no consistent regular CF for integers $a,b,c$ and $d$. – Ng Chung Tak Mar 19 '21 at 15:34
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    Why do you think that it is possible to represent it as a continued fraction? – Somos Mar 19 '21 at 16:04
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    – Ng Chung Tak How could this be proven? – Vertum Mar 19 '21 at 16:52
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    Using Mathematica function “ContinuedFraction” on the 1st expression and varying $a,b,c,d$, the length of the regular CF is not fixed and $a,b,c,d$ is not necessarily re-appeared on the CF itself. But the 2nd will do. – Ng Chung Tak Mar 19 '21 at 18:31

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