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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.

avggrowth

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Written by meditationatae

May 22, 2016 at 8:36 pm

Posted in History

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