Patterns in NS1D0 Sequences
This week, I’m taking a break from looking at the generation of NS1D0 sequences to take a look at some interesting patterns I found within them, patterns that can potentially be used to form new sequences.
Below is an image of a spreadsheet of values. This spreadsheet denotes how often each potential value in an order 17 NS1D0 sequence appears in each position. The rows represent are the numbers, while the columns are the positions. For instance, across all order 17 NS1D0 sequences, a 5 appears in position 7 128 times.
There are some interesting things to note here. The most obvious patterns arise simply from the definition of an NS1D0 sequence; a 0 is always in the first position and a 1 is always in the last, while a 9 never appears.
However, something that is not obvious from the definition of NS1D0 sequences is the presence of rotational symmetry. The chart of [2,1] to [8,7] is a 180 degree rotation of the chart from [10, 1] to [16, 7]; i.e. a 2 appears in position 1 exactly as often as a 16 appears in position 7, a 4 appears in position 5 exactly as often as a 14 appears in position 3, etc. To me, this suggested that there must be a way to translate the numbers above the 9 to the numbers below the 9 in sequences, and as it turns out there is. The following method takes any NS1D0 sequence and transforms it into what I will call its complement (I will illustrate this method using the valid NS1D0 sequence [0, 11, 6, 15, 5, 8, 4, 2, 1]:
1. Reverse the order of the numbers [0, 11, 6, 15, 5, 8, 4, 2, 1] => [1, 2, 4, 8, 5, 15, 6, 11, 0]
2. Change all non-0, non-1 numbers to the number equidistant from the ½ number on the other side (e.g. 3 becomes 15, 14 becomes 4, 7 becomes 11, etc) [1, 2, 4, 8, 5, 15, 6, 11, 0] => [1, 16, 14, 10, 13, 3, 12, 7, 0]
3. Swap the positions of the 0 and the 1 [1, 16, 14, 10, 13, 3, 12, 7, 0] => [0, 16, 14, 10, 13, 3, 12, 7, 1]
As you can see, this process creates another valid NS1D0 sequence.
However, this is not the only pattern I noticed. Another pattern is that on either side of the dividing line of 9’s, the frequencies of each of the numbers follows one of these 3 patterns:
1. They have the exact same frequencies as another number on this side of the divide (e.g. 3 and 4)
2. They have an exactly reversed frequency as another number on this side of the divide (e.g. 2 and 6)
3. They are a palindrome (same frequency in first position as last, second position as penultimate, etc, e.g. 7).
Each of these patterns also suggested to me that a transformation should be possible. Through analyzing the numbers, I was able to find the following pattern for a transformation from a sequence with 3 in the 1 position to a 4 in the 1 position (illustrated with the valid NS1D0 sequence [0, 3, 10, 4, 13, 11, 6, 2, 1]:
1. Change each number in the sequence according to a specific dictionary of transformations (see below) [0, 3, 10, 4, 13, 11, 6, 2, 1] => [0, 4, 16, 8, 3, 6, 5, 11, 1]
2. Keeping the 0, the 4, and the 1 in the same position, reverse the positions of all other numbers (what was the number in the 2 position moves to the 7 position, what was in position 3 moves to the 6 position, etc.) [0, 4, 16, 8, 3, 6, 5, 11, 1] => [0, 4, 11, 5, 6, 3, 8, 16, 1]
The dictionary for how each number transforms looks like this:
{2: 11,
3: 4,
4: 8,
5: 15,
6: 5,
7: 12,
8: 2,
9: 9,
10: 16,
11: 6,
12: 13,
13: 3,
14: 10,
15: 14,
16: 7}
This transformation is far less widely applicable than the complement transformation; it specifically works only on sequences that start with a 3. However, it is indicative of a bigger group of similar transformations. For instance, this transformation will turn a valid sequence that begins with a 7 into a valid sequence that ends with a 7 (Illustrated on [0, 7, 5, 8, 2, 15, 14, 6, 1]):
1. Change each number according to the specific dictionary. [0, 7, 5, 8, 2, 15, 14, 6, 1] => [0, 7, 8, 13, 3, 2, 6, 4, 1]
2. Move each non-0, non-1 number one position to the left, with the 7 moving from position 1 to position 7. [0, 7, 8, 13, 3, 2, 6, 4, 1] => [0, 8, 13, 3, 2, 6, 4, 7, 1].
The dictionary for this transformation is:
{2:3,
3:16,
4:12,
5:8,
6:4,
7:7,
8:13,
9:9,
10:5,
11:11,
12:14,
13:10,
14:6,
15:2,
16:15
}
There are a number of these transformations that change the numbers according to a specific dictionary (with no pattern that I’ve been able to decipher) and then a change in the position of the numbers. I’ve written code to identify each such pattern, and found a number of them in the n=17 ns1d0s. Interestingly, although the occurrences would make it appear that likely a sequence with a 3 in the 2 position could be translated into a sequence with a 4 in the 2 position, there is no transformation of this number translation -> index translation type. In fact, all identified transformations of this type transform sequences that have a specific number in a first or last position to sequences that have a specific number (the same or different) in a first or last position. I suspect this means that there are transformations in other forms that will provide a way to connect those sequences.
Notes on N=15 NS1D0 Sequences
Due to how I was able to discover these patterns, I’ve only been working with NS1D0 sequences of orders 9, 11, 13, 15, and 17. This is because these are the numbers that I can generate the full list of sequences for on my computer. For every order OTHER THAN 15, the occurrences of the numbers all match one of the patterns listed above (repeated here for convenience):
1. They have the exact same frequencies as another number on this side of the divide (e.g. 3 and 4)
2. They have an exactly reversed frequency as another number on this side of the divide (e.g. 2 and 6)
3. They are a palindrome (same frequency in first position as last, second position as penultimate, etc, e.g. 7).
But for order 15, only the pairs 4,7 and 9, 12 fit any of these rules. It is unclear at this point in time if these rules spring from coincidence, and just happen to fit every number in the other order NS1D0 sequences, or if there is a specific reason for them to not appear in the N=15 occurrence chart. However, these pairs do still have the expected possible transformations: 4 in the 1 position to 7 in the 1 position, 4 in the 6 position to 7 in the 6 position, 9 in the 1 position to 12 in the 1 position, 9 in the 6 position to 12 in the 6 position as well as their opposites. Hopefully, we will soon be able to get a full list of order n=19 NS1D0 sequences which may shed some light on whether 15 is an anomaly or a common occurrence.
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