Pisot Sequences

Thanks to the number associated with my diurnal age today, 27204, I was introduced to so-called Pisot sequences that I'm still coming to terms with. Let's start with one of the properties of this number, namely its membership in OEIS A048589:

A048589

Pisot sequence L(7, 9).

The members of the sequence are derived using the following formula:

$$a(n)= \Biggl \lceil \frac{a(n - 1)^2}{a(n - 2)} \Biggr \rceil \\ \text{ where } n \geq 2 \text{ with }a(0)=7 \text{ and } a(1)=9$$The initial members of the sequence are:

7, 9, 12, 16, 22, 31, 44, 63, 91, 132, 192, 280, 409, 598, 875, 1281, 1876, 2748, 4026, 5899, 8644, 12667, 18563, 27204, 39868, 58428, 85629, 125494, 183919, 269545, 395036, 578952, 848494, 1243527, 1822476, 2670967, 3914491, 5736964, 8407928, 12322416, 18059377

However, exactly the same sequence of numbers can be derived from the recurrence relation:$$a(n)=2a(n-1)-  a(n-2) + a(n-3) - a(n-4)\\ \text{ with } a(0)=7, a(1)=9, a(2)=12 \text{ and } a(3)=16$$However, the equivalence may last only up to \(n=1000\) according to the OEIS comments. After that, there is uncertainty. Permalink.

Let's take another example using OEIS A018910:

A018910

Pisot sequence L(4, 5).

Once again, the previous formula can be used with different starting values:$$a(n)= \Biggl \lceil \frac{a(n - 1)^2}{a(n - 2)} \Biggr \rceil \\ \text{ where } n \geq 2 \text{ with }a(0)=4 \text{ and } a(1)=5 $$The initial members of the sequence are:

4, 5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Again, the same sequence can be generated using the recurrence relation (permalink):$$a(n)=2a(n-1)- a(n-3)\\ \text{ with } a(0)=4, a(1)=5 \text{ and } a(2)=7 $$Just to confuse matters we can also generate the sequence using:$$a(n) = \text{ Fibonacci}(n+3)+2$$What the letter L signifies I don't know. Other letters are used with Pisot sequences, including E, P and T. Not all Pisot sequences use the ceiling function. Some use the floor function and others add or subtract a constant. There's not a great deal of introductory information out there about these sorts of sequences. You find yourself in deep mathematical matters very quickly once you start to explore the topic. 

Here is what Google's Bard had to say about them:

Pisot sequences are a type of infinite sequence of real numbers that have a number of interesting and useful properties. They are named after the French mathematician Charles Pisot, who first studied them in the 1930s.

Simple examples of Pisot sequences include:

• The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
• The Lucas sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
• The Parry sequence: 0.1101001000101101...

Pisot sequences are important because they have a number of unique properties. For example:

• They are all irrational numbers.
• They are all aperiodic (meaning that they never repeat themselves exactly).
• They are all dense in the real numbers (meaning that between any two real numbers, there is always a Pisot number).

Pisot sequences are connected to Pisot numbers and Salem numbers in the following way:

• A Pisot number is a real number that is greater than 1 and has the property that all of its conjugates are less than 1.
• A Salem number is a real number that is greater than 1 and has the property that all of its conjugates are either less than 1 or equal to 1.

Every Pisot number is the limit of a Pisot sequence, and every Pisot sequence converges to a Pisot number. Salem numbers are a special type of Pisot number.

Here are some of the applications of Pisot sequences:

• They can be used to generate random numbers.
• They can be used to model complex systems, such as the growth of populations and the spread of diseases.
• They can be used to design efficient algorithms for tasks such as sorting and searching.
• Pisot sequences are a fascinating and important area of mathematics, and they have a wide range of applications in other fields.