Why does $\frac{\pi}{3}e$ contain so many repeating digits?

Question

$\frac{\pi}{3}e$ is approximately equal to:

2.84657807422452235515451695651552483167829617858837165395986704339307620371911026919085462323682464797125831417055915588210706253023200799687278781793023471514007199829654447617582233348895922031024237096797835826800351909013449542434666218846881388726409761904569405458137658722823724181557439473446917219291006835569333157358766322654791237966966946623595073544349432678461546724384333992745425590086011913751990685057546632872448319697537470763226921527595140181054405265139836753917775379775697089746424785856104245156874061419780994637730604531480206814256732829792410814871969646790020050463395548624913283274551800217089819065169939172989203287463740049502718515202660522891769528913026965548612122163723732564317175312804182112811958680271600871936194429224533157202322236063849980512815661793560779711542728888881905074320...

While $e$ contains a lot of repeating digits - I'd normally expect this to disappear once $\pi$ is introduced.

It almost looks like the further along you go, the greater the degree of repetition.

Is this just due to an interesting interaction that occurs only with decimal approximations of pi and e, or is there more going on?

Answer 1

Considering the fractional part shown, there are $77$ occurrences of repeated digits (a digit followed by the same digit) in a string of $830$ digits. The expected number of repeated digits would be $82.9$ with a $\sigma$ of $8.64$. This is $0.683\sigma$ below the mean.

Correction

The search I performed did not count all repeated digits, as I had thought. It started the next search after the end of the previous search, so it missed the second pair that occurs in a triple. The count of doubles is actually $86$. This is $0.359\sigma$ above the mean. Not terribly significant, but above the expected number.