Recursive formulas for the curvature of circles in an Apollonian gasket I was reading Wikipedia's "Apollonian gasket" article and came across this picture:

Over here, it's mentioned that:

The absolute values of the curvatures of the "$a$" disks obey the recurrence relation $$a(n) = 4\,a(n − 1) − a(n − 2)$$

I am trying to better understand this recursive formula:

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*Does have a reference for this formula? I tried to search "absolute value of curvature" and found formulas that did not match (e.g. Wikipedia's "Total absolute curvature" article)


*Does anyone know what "$n$" stands for in this formula? I understand that in a recursive formula, "$n$" usually denotes the index (e.g. in this formula, $a(5) = 4\,a(4) - a(3)$ ) - but  does "$n$" actually refer to the "order of the circles in which they are drawn"? E.g. After drawing the "container" that will contain all future circles -  the first circle we draw is $n = 1$, the second circle we draw is $n= 2$, etc. ?


*How do we know the values for the initial terms in the recursive series? E.g. $a(0)$, $a(1)$, and $a(2)$


*And what exactly do these numbers mean? How is the circle with the number "$32$" different from the circle with the number "$65$"? I understand that the number written on each circle corresponds to the "absolute curvature of that circle" - based on this picture, it seems like circles with "smaller values of absolute curvature" appear larger in size.
Can someone please help me understand the details of this recursive formula?
Thanks!
 A: It might be more clear to write the sequence in subscript notation:
$$a_n = 4a_{n-1} - a_{n-2}$$
Starting with the most simple question:


*

*And what exactly do these numbers mean? How is the circle with the number "$32$" different from the circle with the number "$65$"? I understand that the number written on each circle corresponds to the "absolute curvature of that circle" - based on this picture, it seems like circles with "smaller values of absolute curvature" appear larger in size.


The curvature of a circle is the inverse of its radius. The smaller the circle, the more tightly it has to turn, so the curvature goes up as the radius goes down.


*

*Does anyone know what "$n$" stands for in this formula? I understand that in a recursive formula, "$n$" usually denotes the index (e.g. in this formula, $a(5) = 4\,a(4) - a(3)$ ) - but  does "$n$" actually refer to the "order of the circles in which they are drawn"? E.g. After drawing the "container" that will contain all future circles - the first circle we draw is $n = 1$, the second circle we draw is $n= 2$, etc. ?


You appear to be seriously misunderstanding the table. Each row is for a different Appolonian gasket. The $a$ entries are the curvature of the outside bounding circle, negative because the other circles are inside it. $b, c, d$ are the next three largest circles inside. The $b,c,d$ circles are usually the first three drawn in a gasket. All the other circles, including $a$, are then completely determined by these first $3$.
The table is listing the smallest gaskets that can be formed where the $a, b, c, d$ curvatures are all integer, and two of $b,c,d$ are equal, while the other differs from them by only $1$.
$n$ cannot be "order of circles drawn" because each $n$ represents an entirely different gasket. Evidently, the gaskets are ordered by the size of $|a|$.


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*How do we know the values for the initial terms in the recursive series? E.g. $a(0)$, $a(1)$, and $a(2)$

The entries are the integer solutions to the formula of Descartes' Theorem with $k_2,k_3,k_4$ differing from each other by at most $1$. Calling the common value $m$, and the negative value $a < 0$, this reduces to $$(3m \pm 1 +a)^2 = 3m^2 \pm 2m +1 +a^2$$where the two $\pm$ must agree in sign ($+$ when the third circle is $1$ larger than the other two, and $-$ when it is smaller). Finding the solutions with the smallest values of $a$ is easy enough to do by brute force. Finding the recurrence relation between these $a$ in order of their size likely requires some finesse.


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*Does have a reference for this formula?


Sorry, I do not. But the article gives four, and a number of external links that will likely be helpful. Also, the OEIS reference on the recursion formula in the article itself may be useful - though I do not find this particular instance in which the recursion formula arises mentioned anywhere in the OEIS article.
