Number of sequences as a function of length

Here are the number of valid sequences of short length satisfying C=2.

Could one cook up an explicit (possibly recursive) formula? How does that compare to the Erathosthene sieve: can we see a valid sequence as an analog of a prime number, and if so can we adapt a proof (say Euclid’s) that there are infinitely many primes to this context?

Since any opposite of a valid sequence is another valid sequence, I’ll list only the first half, the total number N being twice that.

UPDATE: corrected some errors!    Now I think I have the definition of discrepancy right:  for a sequence of length L we want, for any given 1\leq n\leq L and for any d=1,\dots ,n-1,  that | \sum_{i=1}^{\lfloor \frac{n}{d}\rfloor} x_{di}|\leq C.

UPDATE: removed last value which was wrong, much more info on the wiki anyway.

L=1: N=2
+

L=2: N=2
++
+-

L=3: N=2
++-
+-+
-++

L=4: N=10
+ + – +
+ – + +
– + + +
+ + – –
+ – + –
– + + –

L=5: N=14
+ + – + –
+ – + + –
– + + + –
+ + – – +
+ – + – +
– + + – +
+ – – + +
– + – + +
– – + + +

Advertisements

Tags:

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: