On counts of finite sequences with discrepancy <= 2
In sci.math, Richard Tobin provided the number of essentially different sequences of discrepancy at most 2, for lengths n from 1 to 75 inclusively.
This leads to a function count(n) , for n = 1 … 75, which is the number of discrepancy D <= 2 sequences of length ‘n’, up to a change of sign.
The growth is very approximately exponential.
Therefore, it’s suitable to define:
f(n) := log( count(n) ).
For example, count(1) = 1, and f(1) = 0.
We now plot the growth, i.e. f(n), minus (the average growth at ‘n’).
This leads to defining:
g(n):= f(n) – C*(n-1) ,
where C is the unique real number such that:
f(75) – C*(75-1) = 0.
Note that we also have g(1) = 0.