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$\gamma$, the Euler-Mascheroni constant, has the following simple regular continued fraction:

$$\gamma=[0; 1, 1, 2, 1, 2, 1,\dots]=0+ \cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 2+\cfrac{1}{ 1+\ddots } } } }$$

I wondered which would be its negative continued fraction, but after searching I have not found anything about it in the literature I have checked.

Just using rough approximation methods, I have derived that it seems that

$$\gamma=1- \cfrac{1}{ 3-\cfrac{1}{ 2-\cfrac{1}{ 3-\cfrac{1}{ 2-\ddots } } } }$$

(although there is an initial pattern in the alternating $2$ and $3$, this pattern ceases).

I would like to know if there is some way to transform a regular continued fraction into a negative continued fraction, to see if that transformation could give some valuable info about $\gamma$. Also, if the negative continued fraction of $\gamma$ is already known, it would be great if you could provide where I can consult it.

Thanks in advance!

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I found this interesting research that I share with you in case you are interested in the relationship between the negative and positive continued fractions:

https://scholarship.claremont.edu/cgi/viewcontent.cgi?filename=2&article=1183&context=hmc_theses&type=additional

According to Definition 8, the first terms of the negative continued fraction of the Euler Mascheroni constant would be:

$$\gamma=[1,3,2,3,2,3,2,2,2,5,2,2,2,2,2,2,2,2,2,2,2,2,7,3,2,2,2,2,2,2,2,3,2,...]$$

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