Other/Special Sequences and Series

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This page originally created by Andy L and Sajid H. (2021)

A sequence in mathematics is a grouping of numbers where the order is important. They often follow a pattern that is repeated within the sequence and can range from simple ones such as the natural numbers sequence


to complex ones like the prime number sequence



These sequences contain terms or elements which refer to the distinct numbers that are a part of the sequence. The amount of terms within a sequence is considered the length and can range from extending infinitely or having a finite length.

When the mathematical sequences have a unique pattern to them, they are referred to as special sequences. One such example of this includes the Fibonacci sequence which is known around the world and is one of the most famous examples of a special sequence. Other lesser-known examples are also quite prevalent in higher mathematics and include such sequences as the Lucas sequences, which is a grouping of constant recursive integer sequences. What this means is that these are sequences that can be continued infinitely once one or more initial terms are known in the sequence. Both the Fibonacci numbers and Lucas numbers are considered Lucas Sequences.

Fibonnaci Sequence[edit]

The design of the Fibonacci Sequence.

The Fibonacci Sequence is a mathematical sequence in which each succeeding number is the sum of the two previous numbers in the sequence. It can be represented as such




Using this equation, the first 12 numbers in the sequence can be determined as follows

Fibonacci numbers.png

This sequence is one of the most well known around the world and can be seen in a variety of different environmental settings. One such example includes the spiral formation of sunflower seeds which, if counted, will result in the Fibonacci sequence. Another example can be seen in leaf arrangements of certain plants such as succulents.

The use of the sequence can also be applied to areas such as computer algorithms where it is used in searching through data and generating graphs.

As mentioned previously, the Fibonacci sequence is a recurrence sequence meaning that succeeding terms can be determined using the previous terms in the sequence. For this reason, the Fibonacci sequence is an infinite sequence as previous terms can be continuously used to find new terms. This would also make a

Look and Say Sequence[edit]

The lines represent the increase in the number of digits in the look and say sequence where the start for each colour is 23 (red), 1 (blue), 13 (violet), 312 (green)

The look and say sequence is a unique sequence due to the method by which the terms are found with. To determine the elements of the sequence, one must simply look and say the current number in the sequence to find the next one. The first eight values are displayed below.

Look and say.png

To determine these values, one just needs to follow these steps

  • The first value is just 1
  • For the second value, you look at the first value where you see "one one" to get 11
  • For the third value you look at 11 and read it as "two one" because there are two 1s
  • For the fourth value you look at 21 and read it as "one 2 one 1"
  • For the fifth value you look at 1211 and read it as "one 1, one 2, two 1s"
  • For the sixth value you look at 111221 and read it as "three 1s, two 2s, one 1"

    This sequence will grow indefinitely and will not contain a digit other than 1, 2, or 3

Triangular Numbers Sequence[edit]

A visual representation of the triangular numbers sequence

The triangular number sequence is a sequence of numbers that can be arranged in the shape of an equilateral triangle. Due to this aspect of the sequence, the triangular number sequence is considered a Figurate number alongside cube numbers and also square numbers. This sequence has an infinite length and can grow indefinitely.

The first 12 terms of the sequence are as follows


The formula used for the sequence is as follows:


Where Tn represents the triangle number, n is the number of dots, and k is a positive integer.

It can also be expressed as


Where Tn represents the triangle number, and n represents the nth triangle number.

Triangle numbers also share a close connection with square numbers. The sum of two consecutive triangle numbers is equal to one square number similarly to how the area of a triangle can be found by halving the area of a square.

A sequence of square numbers

The Lazy Caterer's Sequence[edit]

Cuts made in a circle to represent the Lazy Caterer's sequence

The Lazy Caterer's sequence describes the maximum amount of pieces an object of a disk shape can be cut into with a given number of straight cuts. For example, a circle cut evenly using 3 slices that meet in the middle will result in 6 pieces formed. However, if the cuts were not joined at the middle and were instead cut unevenly, there will be 7 pieces instead. It is more formally known as the central polygonal numbers.

The Lazy Caterer's sequence

Lazy Caterer.png

The central polygonal numbers are represented in this sequence:

Caterer Numbers.png

If they look familiar, it's because each number is the same as the corresponding term in the triangle sequence but increased by one. A triangle sequence is considered a Figurate number.

The Lucas Numbers[edit]

The Lucas number spiral

The Lucas numbers are very similar to the Fibonacci numbers. They both follow the same pattern where you take the sum of the previous two numbers to get the next one in the series. Where they differ here is in the starting values. The starting values for the Lucas sequence are L0 = 2 and L1 = 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1 = 1. Below you will see the first 12 values of the Lucas series.

Lucas number.png

Since both the Fibonacci numbers and Lucas numbers form complementary instances of Lucas sequences, they share some relationships with one another. One of these relationships is when you add any two values on the Fibonacci sequence which are two terms apart, you will get the Lucas sequence number that is in-between them.

Another one of these relationships that they share is that they are both recursive, meaning that the next term can be found using previously known terms within the sequence.

While determining the Lucas numbers, one can use terms within the Fibonacci sequence using the equation below because of how closely they are related



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