# How many DIFFERENT DIGITS are contained in a sequence of length L?

## GENERAL CONSIDERATIONS / EXPECTED VALUES

When testing longer chains we gonna start with a simple question:

How many DIFFERENT DIGITS are contained in a chain of length L?

Let us call this number the Variety of digits VOD.

Here is an example for Pi with L=5.

14159 -> VOD = 4 ; 26535 -> VOD = 4 ; 89793 -> VOD = 4 ; 23846 -> VOD = 5

26433 -> VOD = 4 ; 83279 -> VOD = 5 ; 50288 -> VOD = 4; 41971 -> VOD = 4

69399 -> VOD = 3 ; 37510 -> VOD = 5

The construction law for the VOD-distribution is easy to find.

Let w(L,d) be the probability that a chain of length L consists of exactly d different digits.

For L=1 we have even one digit:

w(1,d)=1 for d=1 and w(1,d)=0 for d>1

For L>1 any w(L,d) can be calculated from its predecessor:

w(L+1,d) = w(L,d) x d/10 + w(L,d-1) x (10-d)/10

Finding a VOD=1 for L=5 is just the same as searching for a single digit run of length 5 (e.g. 11111).

But analyzing single digit runs is limited to L~7 due to the available data.

Thus testing variety of digits up to L=40 is a chance to get a better feeling for the regularity of single digit distribution. More than this the VOD can be an indicator for clusters.