Upward continued fractions

Question

Has anybody seen "upward continued fractions", such as $$ \frac{1+\large{\frac{1+\large{\frac{1+...}{a_2}}}{a_1}}}{a_0} \quad? $$

These can be formed, for any real number $x$ with $0<x\le 1$, by defining $x_0:=x$ and inductively $$ a_n:=\lfloor{x_n}\rfloor+1\qquad\text{and}\qquad x_{n+1}:=a_nx_n-1. $$ It is easy to check that $1\le a_1\le a_2\le a_3\le\dots$, and that the sequence of $a_i$'s is eventually constant if and only if $x$ is rational. This procedure yields expressions like the one displayed above, since $$ x = \frac{1+x_1}{a_0}=\frac{1+\large{\frac{1+x_2}{a_1}}}{a_0} =\frac{1+\large{\frac{1+\Large{\frac{1+x_3}{a_2}}}{a_1}}}{a_0} = \cdots $$ I looked for these in google, wikipedia, and standard texts like Perron's, but couldn't find them. Have they been studied? I ask because a high school student invented these before my eyes today, and I'd like to tell him what he's rediscovered (assuming that in fact these have been studied before).

Answer 1

Expanding my comments into an answer: by distributing the divisions by $a_0$, $a_1$, $\ldots$ successively you can rewrite such an upward continued fraction in the equivalent form $\frac1{a_0}+\frac1{a_0a_1}+\frac1{a_0a_1a_2}+\cdots$. This is known as the Engel expansion of the number, and their coefficients have some interesting limiting properties (in particular, for almost all real numbers the coefficients grow exponentially); the Wikipedia article on them should offer up several good pointers for more information.

Answer 2

You can find plenty on information on those special continued fractions at the following link (American Mathematical Monthly):

https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/1055084257873891/?type=3&theater

Also at: https://youtu.be/mjSHenvXsEs

Answer 3

Five years ago or so I worked on some "tree-type" continued fractions... $$ T = 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\frac{1+\cdots\;}{2+\cdots\;} }{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} } }{\displaystyle 2+\frac{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} }{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} } } }{\displaystyle 2+\frac{\displaystyle 2+\frac{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} }{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} } }{\displaystyle 4+\frac{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} }{\displaystyle 8+\frac{8+\cdots\;}{16+\cdots\;} } } } \;. $$

But it never went very far. I did submit one of these as a problem... Math. Mag. 79 (2006) p.151