Exact value for the continued fraction of $\tiny 1+\cfrac{1}{3+\cfrac{3}{5+\cfrac{5}{7+\cfrac{7}{9...}}}}$?

added
Comments point out that this post did the wrong continued fraction. For the correct one, use $a=-1$ not $1$. Then follow the same Satz $2$. The result is $$ \frac{2\;{}_2F_1(-\frac12;1;\frac12)}{{}_1F_1(\frac12;2;\frac12)} =\frac{I_0(\frac14)+I_1(\frac14)}{I_0(\frac14)-I_1(\frac14)} \approx 1.2831923 . $$

original post
Here is the reference for everything on continued fractions (as of 1913):

Perron, Oskar, Die Lehre von den Kettenbrüchen., Leipzig - Berlin: B. G. Teubner. xiii, 520 S. $8^\circ$ (1913). ZBL43.0283.04.

Section 81, Satz 2 evaluates

$$ c + \frac{a+b}{\displaystyle c+d + \frac{a+2b}{\displaystyle c+2d+\frac{a+3b}{\displaystyle c+3d+\ddots}}} $$

So we need $a=1,b=2,c=1,d=2$.

Value of the continued fraction is $$ \frac{2\;{}_1F_1(\frac12, 1, \frac12)}{{}_1F_1(\frac32, 2, \frac12)} \approx 1.779306397 $$ This can be written $$ \frac{2e^{1/4} I_0(\frac14)} {e^{1/4} I_0(\frac14)+e^{1/4} I_1(\frac14)} = \frac{2}{\displaystyle 1+\frac{I_1(\frac14)}{I_0(\frac14)}} $$ in terms of Bessel functions.