A Family of Continued Fractions with Closed Forms Generalizing a Conjecture from the Ramanujan Machine

In a previous work, the author proved a conjectured continued fraction identity catalogued by the Ramanujan Machine project. The identity involved the constant $1/(1-\ln 2)$. In this paper we show that the original continued fraction is a special case of a large family that can be evaluated in closed form. We derive a two-parameter family of continued fractions \[ a_0+\cfrac{b_1}{a_1+\cfrac{b_2}{a_2+\cfrac{b_3}{a_3+\cdots}}}, \] with \[ a_n=\Bigl(\frac{a_0}{2}+t\Bigr)n^2+\Bigl(\frac{3a_0}{2}+t\Bigr)n+a_0,\qquad b_n=-\frac{a_0 t}{2}\,n^2(n+1)^2, \] and prove that its limit is \[ \frac{a_0 r^2}{2\bigl(r+(1-r)\ln(1-r)\bigr)},\qquad r=\frac{2t}{a_0}. \] The proof uses an explicit closed form for the numerator sequence $P_n$, a telescoping difference equation, and elementary integration. We also discuss the analytic continuation of the formula to complex parameters and outline higher-order analogues that lead to combinations of logarithms and rational functions.