The terms **sequence** and **series** sound very similar, but they are quite different. Basically:

- A
**sequence**is a set of ordered numbers, like 1, 2, 3, …, - A
**series**is the*sum*of a set of numbers, like 1 + 2 + 3….

**Sequence and Series: Contents:**

- Sequence Definition
- Sequence Rules
**Types of Sequences**:- Arithmetic Progression
- Arithmetic Sequence
- Bounded Sequence
- Cauchy Sequence
- Complementary Sequence
- Complete Sequence
- Constant Sequence, Eventually Constant
- Ducci Sequence
- Exponential Sequence
- Fibonacci Sequence
- Finite Geometric Sequence
- Finite Sequence
- Generating Sequence / Function
- Increasing Sequence
- Infinite Sequence
- Infinite Geometric Sequence
- Integer Sequence
- Monotonic Sequence / Series
- n-tuple
- Numeric Sequence
- Periodic Sequence, Purely / Ultimately
- Polynomial Sequence
- Quadratic Sequence
- Random Sequence
- Recamán Sequence
- Recursive Definition of a Sequence
- Sequence of Partial Sums
- Stationary Sequence: Weakly/Strongly
- Universal Sequence

- What is a Series?
**Types of Series**:- Alternating Series
- Arctangent Series Expansion
- Asymptotic Series / Poincaré Expansion
- Binomial Series
- Continuous Series & Discrete Series
- Dirichlet Series
- Divergent Series
- Finite Series
- Fractional Power Series: Definition, Examples
- Fourier Series
- Geometric Series
- Harmonic Series, Alternating Harmonic Series
- Infinite Series
- Laurent Series
- Monotonic Sequence / Series
- Mercator Series
- Oscillating Series
- P-Series
- Positive Series
- Power Series
- Stirling Series
- Telescoping Series
- Trigonometric Series

**Related articles: **

- Asymptotic Error (Rate of Convergence)
- Cauchy Convergence
- Find the Limit of a Sequence
- Monotone Convergence Theorem: Examples, Proof
- Series Expansions.
- Sequence of Functions
- Subsequential Limit
- Sum of a Series: How to Find

## What is a Sequence?

A **sequence **is a collection of elements (usually numbers) with two major differences from plain old “sets”:

- The elements (or terms) are in order.
- The values of the terms can repeat.

The number of elements is called the *length *of the sequence. The length is potentially infinite, meaning it goes on without end. If the collection of numbers ends then it is called a **finite sequence**.

## Sequence Rules and Arithmetic Sequences

Sequences generally have a **rule**. The rule is applied to find the value of unknown terms. For example, the sequence {3, 6, 9, 12…} begins at 3 and increases by 3 for every subsequent value. This is an example of an **arithmetic sequence** with a **common difference** of 3.

The above rule is helpful in defining values in the immediate vicinity of the preceding terms. A limitation of the rule, however, is that it does not tell us the value of an unknown term. In the example above, we don’t know what the 100*th* term will be. In other words, we need a rule that can quickly determine the value for any unknown term.

## Formula and Example for Nth Term

The formula for the *n*th term of the arithmetic sequence (also called the general term) is:

**A _{n} = a + (n -1)d**

Where:

- a
_{n}is the nth term of the sequence, - a is the first term,
- d is the common difference.

The *n*th term is the unknown term that you are trying to calculate. Using the formula above you can quickly calculate any *n*th term.

**Example question**: Find the 100th term for {3, 6, 9, 12…}:

Step 1: Determine the values for *a*_{n}, *d* and *n*.

- a
_{n}(the nth term that you are trying to find) = 100, - a is the first term: 3,
- d is the common difference (3),

Step 2: Place each value from Step 1 into the formula:

*a*_{100}= 3 + (100 – 1) 3- = 3 + (99) 3
- = 3 + 297

*a*_{100} = 300

The 100th term is 300.

## Arithmetic Progression

An **arithmetic progression** (or arithmetic sequence) is a list of numbers, where every term increases by the same amount—called a common difference. In other words, to go from one term to the next, you just add a number. For example, starting with 1:

- {1, 2, 3} … add 1 each time,
- {1, 7, 14} … add 7 each time.

You don’t have to start with 1 though; You can start with any number. What’s important is that you add the same constant each time.

## Finding Common Difference in Arithmetic Progression

The common difference is how much is added to each term in the sequence. For example, the sequence {2, 4, 6, 8} increases by 2—the common difference. Another example: {1, 25, 49} has a common difference of 24. In notation, the common difference is often written as *d*. For example:

a, a + d, a + 2d, a + 3d, a + 4d.

Which can also be written as a + (n · d), where n starts at zero.

For example, if a = 1 and d = 2:

1, 1 + 2, 1 + 2(2), 1 + 3(2), 1 + 4(2) = 1, 2, 5, 7, 9.

**Example question: **What is the common difference for the arithmetic progression {5, 8, 11, 14, 17}?

Step 1: Identify the first term. In this list, that’s 5.

Step 2: Subtract the first term from the second: 8 – 5 = 3.

The common difference is 3.

## Finite and Infinite Arithmetic Progression

The progression can be a fixed (finite) amount of numbers or an infinite amount. For example, starting with 2 and using a common difference of 3, you get the arithmetic sequence {2, 5, 8, 11…}. The three dots (…) indicate that the sequence goes on an on until infinity. When you have a fixed number of terms, the sequence is called an *n-term arithmetic progression*. For example, the sequence {2, 5, 8} is a three term arithmetic sequence.

## Arithmetic Progression Examples from Number Theory

Arithmetic progression is heavily used in number theory—especially in the analysis of prime numbers. A rather more complex example of an arithmetic progression from number theory:

**{a + mk: k >0} ⊆**

Where:

- m = fixed integer > 0 &
- a = fixed integer ≥ 0
- Where:
- a + mk = equals a prime number,
- gcd(a, m) = 1.

- Where:
- ⊆ = subset of (or equal to),
- k = a natural number,
- ℕ = set of natural numbers.

The above example is called **Dirichlet’s theorem on primes in arithmetic progressions** (Lozano-Robledo, 2019). Related to Dirichlet’s theorem are these two important examples of arithmetic progression, which contain all prime numbers except for 2 and 3 (Caldwell, 2020):

- 1, 7, 13, 19, 25, 31, 37, …
- 5, 11, 17, 23, 29, 35, 41, …

## Complementary Sequence

A **complementary sequence** is the inverse of a strictly increasing sequence of non-negative integers (the counting numbers, 1, 2, 3…). “Inverse” refers to the fact it contains numbers not in the original sequence.

More specifically, if a natural-numbered sequence (a_{n}) is a strictly increasing sequence of non-negative integers then the complementary sequence is **the set of all non-negative integers which are not terms of the sequence** [1].

A sequence of positive integers and its complement contain all possible positive integers.

## Example of a Complementary Sequence

The complementary sequence of even non-negative integers {2, 4, 6, 8, …} is the odd non-negative integer sequence {1, 3, 5, 7, …}.

This also works in reverse: The complementary sequence of {1, 3, 5, 7, …} is the sequence {2, 4, 6, 8, …}.

## Complementary Sequence in Biology

The definition of a complementary sequence in biology is slightly different than the mathematical definition, but the principal is the same. In a way, you can think of them as *opposites*.

**DNA sequencing.**DNA is double-stranded; For each strand with a certain sequence, there is another strand that complements it (called the

*Watson-Crick pairing rule*). For each strand of DNA with a given sequence, there is a complementary one where Adenine (A) complements Thymine (T) and Guanine (G) complements Cytosine (C) [4].

There is a similar complementarity with RNA: Adenine complements Uracil and Guanine complements Cytosine [5].

A pair of **Golay complementary sequences ** is a pair (a, b) of sequences with out-of-phase (i.e. shifts) autocorrelations equal to zero. In other words, C_{a}(u) + C_{b}(u) = 0, 0 < u < n [2]. For example, (1 1 -1 1) and (1 1 1 -1) are a Golay complementary pair [3]. These sequences have many applications, including infrared multislit spectrometry and Orthogonal Frequency-Division Multiplexing.

## References

[1] Mortich, S. (2010). Remarks on Complementary Sequences. Retrieved April 10, 2021 from: https://www.fq.math.ca/Papers1/48-4/Mortici.pdf

[2] Fieldler, F. & Jedwab, J. (2006). How Do More Golay Sequences Arise? Retrieved April 10, 2021 from: https://www.sfu.ca/~jed/Papers/Fiedler%20Jedwab.%20More%20Golay%20Sequences.%202006.pdf

[3] Kalashnikov, E. (2014). An Introduction to Golay Complementary Sequences. Retrieved April 10 2021 from:https://journals.library.ualberta.ca/eureka/index.php/eureka/article/view/22829

[4] Wellcome Genome Campus Advanced Courses and Scientific Conferences. Grammatical rules for DNA sequence representation. https://www.futurelearn.com/info/courses/bacterial-genomes-bioinformatics/0/steps/47002

[5] Laventa, R. & Cesarini, G. (1981). Base pairing of RNA I with its complementary sequence in the primer precursor inhibits ColE1 replication. Nature 294. 623-626.

Image: Zephyris, CC BY-SA 3.0

## Complete Sequence

An increasing sequence of positive integers is a **complete sequence** if every term can be written as a sum of the first term, using the first term at most once [1].

A couple of variations on this definition:

- Schissel [2] states that a complete sequence if every positive integer is a sum of “one or more distinct terms” in the sequence.
- Erdos & Graham [3] describe a complete sequence as one where “every

*sufficiently large*” natural number is a sum of distinct terms of the sequence. This particular definition is called a**weakly complete sequence**[4]. - Linz & Jones [5] define a
**r-complete sequence**, where every sufficiently large positive integer can be represented as the sum of*r or more distinct terms*from the sequence.

## Complete Sequence Examples

A simple example of a complete sequence is {1, 2, 3, 4, …}.

Some complete sequences are more challenging to spot. For example, {1, 2, 3, 4, 8, 12, 16, 20, 24, 28, …} meets the definition because we can represent positive integers in modulo 4 (an arbitrary positive integer, a, can always be written as *a* = *n* * *q* + *r*).

The prime numbers are a complete sequence (if you add 1).

The Fibonacci sequence {1, 1, 2, 3, 5, 8, …} is an example of a complete sequence. Here [2],

- f
_{l, 1}(1) = 1, f_{1, 1}(2) = 1, and - f
_{1, 1}(n) = f_{1, 1}(n – 1) + f_{1, 1}(n – 2) if n ≥ 3.

Removing a single number still leaves a complete sequence, although removing two numbers does not [6].

## Complete Sequence: References

[1] Earl, R. (2017). Towards Higher Mathematics: A Companion. Cambridge University Press.

[2] Schissel , E. (!987). Characterizations of Three Types of Completeness. Retrieved April 7, 2021 from: https://www.fq.math.ca/Scanned/27-5/schissel.pdf

[3] Erdos, P. & Graham, R. (1980). Old and New Problems and Results in Combinatorial Number Theory: Monographie Numero 28 de L’Enseignement Math&matique .

Lf Enseignement Mathematique de 1’Universite de Geneve.

[4] Fox, A. & Knapp, M. (2013). A Note on Weakly Complete Sequences. Journal of Integer Sequences.

[5] Linz, W. & Jones, E. (2016). r-Completeness of Sequences of Positive Integers. Retrieved April 7, 2021 from: https://www.emis.de/journals/INTEGERS/papers/q59/q59.pdf

[6]Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.

## Ducci Sequence

A **Ducci sequence** (or *N-number game*) is a sequence of * n-tuples* (i.e. a list of integers). Each term in the sequence is formed by finding the absolute difference of neighboring integers [1].

## Ducci Sequence Example

**Example question**: Create a Ducci sequence for the *n*-tuple (8, 11, 2, 7).

Step 1: **Label the n-tuple’s terms a to n**:

- a = 8
- b = 11
- c = 2
- d = 7.

Step 2: **Subtract pairs of terms (a and b, b and c, c and d …). **The last term is paired with the first time, so in this example d will be subtracted from a:

- a – b = 8 – 11 = -3
- b – c = 11 – 2 = 9
- c – d = 2 – 7 = -5
- d – a = 7 – 8 = 1.

Step 3: **Place the terms from Step 2 into a new n-tuple, **ignoring the negative signs (by taking the absolute value):

(3, 9, 5, 1).

Step 4: **Connect the new n-tuples **from Step 3 with the Ducci sequence so far. We started with (8, 11, 2, 7), so:

(8, 11, 2, 7) → (3, 9, 5, 1).

Step 5: **Repeat Steps 1 to 4 **until the sequence wither reaches an n-tuple of all zeros or a periodic (repeating loop):

(8, 11, 2, 7) → (3, 9, 5, 1) → (6, 4, 4, 2) → (2, 0, 2, 4) → (2, 2, 2, 1) → (0, 0, 0, 0).

*That’s it!*

## A Graphical Ducci Sequence

The Ducci sequence can also be calculated using squares or circles, although you should start with a large enough shape (perhaps a whole sheet of paper) so that you don’t run out of room.

**Example #2:** Create a graph for the Ducci sequence (6, 9, 0, 5)

Step 1: **Draw a circle** and, starting at the top, write the sequence in a clockwise direction.

I’ve highlighted the first term of each iteration in red, so that it will be easier to reconstruct the sequence when we’ve finished.

Step 2: **Subtract pairs of terms**, starting with the red term and moving in a clockwise direction. Write down the absolute value of each difference:

Make sure to highlight the first term, which is 6 – 9 = 3 in this example.

Step 3: **Complete the sequence,** repeating Step 2 until you get all zeros or have a periodic sequence:

## Ducci Sequence: References

[1] 4723 Ducci Sequence. Retrieved May 5, 2021 from: https://icpcarchive.ecs.baylor.edu/external/47/4723.pdf

## What is a Generating Sequence?

A **generating sequence** (also called a *generating function*) is one way to create a finite sequence.

For example:

- The generating sequence a
_{n}= c_{n}* r^{n}results in the geometric series if the c_{n}s are constant [1]. - The sequence {6, 26, 66} is generated by the formula [x(x
^{2}+ 4x + 1)].

The most important reason for finding the generating function for a sequence is that functions have a much larger “toolbox” to work with. For example, you can’t find derivatives and integrals of sequences, but you can apply those procedures to functions.

**Not all sequences have generating sequences.** For example, it’s not possible explicitly generate an infinite sequence, but you can generate one for a part of it by using a partial sum [2]. For example, the nth partial sum of the generating sequence a_{n} is [3]:

## Ordinary Generating Function

The **ordinary generating function** specifically refers to a formal power series, where the coefficients correspond to a sequence. The general form is:

Which can also be written as:

G(x) = g_{0} + g_{1}x + g_{2}x^{2} + g_{3}x^{3} + ….

The ordinary generating function is a “formal” power series because the x is used as a placeholder instead of a number; In rare circumstances you might put a value in for x, but leaving the placeholder in means that you can ignore the issue of convergence. [4] It is primarily used in computer science and mathematical analysis.

## Other Meanings of “Generating Function”

The term “generating function” is loosely defined. Sometimes it’s used as a synonym for the generating sequence described above. To avoid this confusion, the generating function that specifically generates sequences is sometimes called a *function for generating sequences *.

Alternatively, it could refer to a specific type of computational tool. For example:

- Moment Generating Functions (MGFs) are an alternative way to represent probability distributions; Each distribution has a unique MGF. They are used to find moments like the mean(μ) and variance(σ
^{2}). - Probability Generating Functions are very similar to MGFs and contain the same information. However, the PGF is usually concerned with non-negative, integer-valued random variables.
- A Cumulant Generating Function (CGF) takes the moment of a numerical sequence.

## Generating Function: References

[1] Chapter 2: Limits of Sequences.

[2] Basic Calculus Concepts.

[3] Limits of Sequences.

[4] Meyer, A. & Rubinfeld, R. (2005). Generating Functions. Retrieved April 4, 2021 from: https://www.math.cmu.edu/~lohp/docs/math/2011-228/mit-ocw-generating-func.pdf

## Increasing Sequence

An **increasing sequence** increases as you travel along the number line.

In notation:

- a
_{1}≤ a_{2}… or - a
_{n}≤ a_{n+1}[1].

In other words, the second term is larger than or equal to the first term, the third term is larger or equal to the second, and so on.

The above definition poses a small problem in that it can define a sequence where the numbers are all equal to each other. When this happens, it’s usually called a constant sequence instead. In order to be a true “increasing sequence”, at least one of the terms in the sequence has to be larger than the one before it. For example:

- Constant sequence: {1, 1, 1, 1, 1}
- Increasing sequence: {1, 1, 1, 1, 2}

Some authors use the term *non-decreasing sequence* to describe an increasing sequence and *strictly increasing* to describe increasing sequences. This may be more intuitive, because a constant sequence is certainly non-decreasing. However, all the different terminology can get a little confusing; Make sure you understand exactly what definition your textbook author (or professor) is using.

## Monotonically Increasing / Strictly Increasing Sequence

At first glance, the definition for a monotonically (or monotone) increasing sequence looks exactly the same as the one for the plain old “increasing” one:

A sequence of natural numbers is monotone increasing if a

_{n}≤ a_{n + 1}[2].

Again, the confusion here is that there really isn’t a difference between the two definitions. Some authors simply use the term “monotonically increasing” to distinguish it from *strictly increasing sequences*. Strictly increasing sequences always increase from term to term; monotonically increasing sequences can stay constant somewhere— they don’t always have to increase. For example:

- Strictly increasing: {1, 2, 3, 4, 5}
- Monotone increasing: {1, 2, 2, 3, 4}

## Ascending and Descending Sequences

If you can stand a little more confusion, non-decreasing sequences are also called *weakly increasing sequences*. Edsger Wybe Dijkstra, one of the most influential members of computing science’s founding generation, suggested that a better replacement for both terms is “ascending”. Imagine climbing (ascending) a staircase. It’s certainly possible to take a breather here and there (stay constant), and you can’t go down (in that case, you would be descending). so Professor Dijkstra’s alternative definition makes a lot more sense.

## Increasing Sequence: References

[1] Zhou, A. Monotone Sequences. Retrieved April 7, 2021 from: https://www.math.ucla.edu/~azhou/teaching/19S/131-week-05.pdf

[2] Goodall, A. Mathematical Analysis I. Retrieved April 7, 2021 from: https://iuuk.mff.cuni.cz/~andrew/MAex4s.pdf

[3] Dijkstra, E. Largely on nomenclature.

https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/ewd768.html

## Integer Sequence

A *sequence* is an ordered list of numbers; An **integer sequence** is an ordered list of integers (whole numbers plus negatives and zero).

There are hundreds of thousands of integer sequences [1] and hundreds of ways to create them. These range from the simple (adding or multiplying by a constant) to the more complex involving transformations, complementary functions or convolutions [2].

## Types of Integer Sequence

Three of the most famous:

**Natural numbers**: The counting numbers (1, 2, 3, …) is the most commonly used integer sequence.**Prime numbers**: whole numbers (numbers that aren’t fractions) greater than 1 that are divisible only by itself and one. { 2, 3, 5, 7, 11, 13}.**Fibonacci numbers**: Every term is the sum of the two before it: (0, 1, 1, 2, 3, 5, 8, 13,…).

And three oddball ones:

- The universal sequence is a sequence that can (theoretically) create every sequence in the universe. The computing power to run such a sequence doesn’t exist (and never will).
**Recamán’s Sequence**: A fun sequence that seems to have no purpose other than to confuse with its mysterious outputs.**Constant Sequence**: A sequence that has all of the same numbers. For example, {1, 1, 1, 1, 1, …).

## Uses of Integer Sequence

Integer sequence appear in just about every branch of mathematics and science, including [3]:

- Chemistry (e.g. atom cluster sizes).
- Computer science (e.g. number of steps required to sort x items).
- Enumeration problems (e.g. combinatorics, graph theory, lattices).
- Number theory (for example, a list of solutions to x
^{2}+ y^{2}+ z^{2}?). - Physics (e.g. paths on lattices).

## Who Invented the First Integer Sequence?

The earliest mathematical artifact known to man is the Lebombo bone (c. 33, 000) B.C., which has 29 tally marks; It is the oldest sequence listed in the OEIS. The British Museum has an old Babylonian clay cuneiform tablet, containing a table of squares and cubes [4]. But our knowledge is limited to those artifacts that are in existence: most are lost to time.

The real question is, who created the integers? Because once they were created, the sequence would have fallen into place. Perhaps Stephen Hawking was right when he said that “God Created The Integers” [5] (or perhaps it was his was of saying “We’ll never know”).

## Integer Sequence: References

[1] The Online Encyclopedia of Integer Sequences (OEIS). https://oeis.org/

[2] Khovanova, T. (2007). How to Create a New Integer Sequence. Retrieved April 10, 2021 from: https://arxiv.org/pdf/0712.2244.pdf

[3] Sloane, N. My Favorite Integer Sequences. Retrieved April 10, 2021 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.64.2824&rep=rep1&type=pdf

[4]. The British Museum. Tablet: 92698.

[5] Hawking, S. (2005). God Created the Integers: The Mathematical Breakthroughs that Changed History. Running Press.

## Numeric sequence

A **numeric sequence** is a list of ordered numbers. A couple of simple examples: 1, 2, 3 or 99, 89, 79.

The geometric sequence is a well-known numeric sequence in calculus. Each term in the sequence is found by multiplying (or dividing) by the same value. For example, the sequence 1, 2, 4, 8, 16 is geometric because each number is a result of multiplying the one before it by 2 (e.g. 1 * 2 = 2, 2 * 2 = 4).

The numbers don’t have to be in ascending or descending order to be called a “numeric sequence”. They just have to be in some order than makes sense for the purpose at hand. For example, the following numeric sequence was used to create a histogram [1]:

## Formal Definition of a Numeric Sequence

In calculus, a numeric sequence is usually defined a little more formally. First, the numbers in the sequence must be real numbers.

Second, each term in the sequence must be a result of either adding or subtracting the term before it. Formally, that is stated as:

- s
_{1}: = a_{1} - s
_{2}: = a_{1}+ a_{2} - s
_{3}: = a_{1}+ a_{2}+ a_{3}

Or, it can be more simply represented by the following summation notation:

## Numeric Sequence in Computational Logic

Computational logic uses computers to establish facts in a logical formalism [2]. The facts are defined much more precisely than simple “a sequence of ordered numbers”:

A sequence of characters is a numeric sequence if and only if one of the following is true [3]:

- The sequence is an optionally signed (±) base 10 digit sequence which may or may not have a dot character on the right-hand side.
- The sequence is a #-sequence. A #-sequence has a length of three or more. The first character is always #. The numeric value is the sequence’s base n signed value found by deleting the first two characters. “n” 2, 8, or 16 depending on whether the second character of the sequence is is B, O, or X).

## References

[1] Yakir, B. (2011). Introduction to Statistical Thinking (With R, Without Calculus). Retrieved April 3, 2021 from: https://my.uopeople.edu/pluginfile.php/57436/mod_book/chapter/37629/MATH1280.IntroStat.pdf

[2] Proceedings of the Royal Society. Computational logic: its origins and applications. https://royalsocietypublishing.org/doi/10.1098/rspa.2017.0872

[3] Boyer, R. & Moore, S. A Precise Description of a Computational Logic. Retrieved April 3, 2021 from: https://www.cs.utexas.edu/users/boyer/ftp/cs.389r.course.packet.pdf

## Random Sequence

A **random sequence** is a sequence of numbers with zero correlation. In other words, given one number in a sequence {r_{1}, r_{1}, r_{1}, …) there’s no way to predict the second number.

Mathematicians have never been able to agree what is “random” and what isn’t. Many authors have tried (and failed) to pin down a formal definition. Others doubt if such a definition is possible [1]. Every sequence is going to show some kind of pattern or regularity [2].

## A Random Sequence Isn’t Really Random

Some sequence that you think are random actually aren’t at all. For example, if you toss a coin three times, you might think that getting a HHT or TTT or THT is completely random. But that’s not the case, because tossing a coin (assuming it’s a fair one) actually follows the laws of probability: your odds of getting a heads are ½ and your odds of getting a tails are ½.

**Computerized** “Random Sequence Generators” aren’t truly random either: someone programmed it with a set of deterministic rules that govern what sequence is spat out. Generating a sequence of random number is impossible [3].

What about a **human generated** random sequence? Off the top of my head, I’m going to type a sequence of random numbers from my laptop keypad (I’ll even keep my eyes closed):

{4578656846879}.

Is this an example of a random sequence? Perhaps surprisingly, the answer is no. Even if you were asked to generate a random sequence of 300 digits between 0 to 9, the result wouldn’t be random either. These types number sequences are not mathematically random; the internal random number generator in your brain is subject to bias, always producing certain patterns [4].

## Random Sequence: References

[1] Church, A. On the Concept of Random Sequences. Retrieved 4/9/2021 from: http://www.socsci.uci.edu/~bskyrms/bio/readings/church_randomness.pdf

[2] Random sequences. Retrieved 4/9/2021 from: https://web.northeastern.edu/afeiguin/phys5870/phys5870/node57.html

[3] Volchan, S. What is a Random Sequence? Retrieved 4/9/2021 from: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Volchan46-63.pdf

[4] Schulz, M. et al. (2012) Analysing Humanly Generated Random Number Sequences: A Pattern-Based Approach. Retrieved 4/9/2021 from: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0041531

## Recursive Definition of a Sequence

The word “*recursive*” means to recur or repeat. In mathematics, it’s the repeated application of a function or rule to successive results. A **recursive definition of a sequence** defines terms by previous entries in the sequence. It takes the rule for the sequence and applies it over and over, starting with the first term. For example:

- A
_{0}= 0 - A
_{n + 1}= A_{n}+ n + 1

The first term A_{0} is defined explicitly, as zero. This is the starting point— you need this to build a recursive definition. Every subsequent term in the sequence is defined recursively in terms of A_{n}. In other words, you need two parts for a recursive definition of a sequence:

**The initial term**(e.g. A_{0}= 0),**A recurrence relation (also called a recursion formula):**a symbolic description of subsequent terms (e.g. A_{n + 1}= A_{n}+ n + 1).

The other way to define sequences is with a general term, also called an explicit definition. For example, the sequence defined by a_{n} = 1/n only requires you to plug in a value for n. You do not need to know the initial term in order to solve the sequence.

## Recursive Definition of a Sequence: Examples

The following sequence is defined recursively with an initial term and a rule for subsequent terms:

- A
_{0}= 3 - A
_{n + 1}= A_{n}+ 5.

The first few terms of the sequence are:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5 =**8**, - A
_{2}= A_{1}+ 5 = (3 + 5) + 5 =**13**, - A
_{3}= A_{2}+ 5 = [(3 + 5) + 5] + 5 =**18**, - A
_{4}= A_{3}+ 5 = {[(3 + 5) + 5] + 5} + 5 =**23**.

_{1}and f

_{2}= 1 and the recursion formula f

_{n}= f

_{n-1}+ f

_{n-2}for n ≥ 3. Each term in the sequence (after the first two) is the sum of the previous two terms. The first few terms are: 1, 1, 3, 5, 8, 13, 21,….

## How to Find the Explicit Definition from a Recursive Definition of a Sequence

There isn’t a formula you can follow to turn a recursive definition to an explicit definition of a sequence. However, there are a few general steps that you can use to identify the pattern needed for an explicit definition.

Step 1: Write out the first few terms of the sequence using the recursive definition, without actually carrying out the arithmetic operations. Using the example above:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5, - A
_{2}= A_{1}+ 5 = (3 + 5) + 5, - A
_{3}= A_{2}+ 5 = [(3 + 5) + 5] + 5, - A
_{4}= A_{3}+ 5 = {[(3 + 5) + 5] + 5} + 5.

Step 2: Look for ways to combine terms. A quick glance at the above terms tells us that 5 is repeating, so we can combine them to get:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5 - A
_{2}= A_{1}+ 5 = 3 + 2(5), - A
_{3}= A_{2}+ 5 = 3 + 3(5), - A
_{4}= A_{3}+ 5 = 3 + 4(5).

From here, we can see a pattern emerge: each step in the process multiplies 5 by the term’s index (the index is the subscript in A_{n}). For example, the third (A_{3}) has 3 * 5. That leads to the general formula:

**a _{n} = 3 + 5n.**

## Recursive Definition of a Sequence: References

Lameda, B. & Nikolaev, N. (2016). Integral Calculus. Retrieved February 4, 2021 from: http://www.math.toronto.edu/beatriz/files/MAT137/MAT136_Lecture_Notes.pdf

## What is a Recamán sequence?

There are **two different sequences** attributed to Columbian puzzle maker Bernardo Recamán Santos:

**Recamán sequence #1** (entry A5132 in the OEIS) is usually the one referred to as “The” Recamán Sequence. It is defined by [1]:

- a
_{n}= a_{n-1}– n, if a_{n-1}– n > 0*and*a_{n-1}has already occurred in the sequence. - Otherwise, a
_{n}= a_{n-1}+ n.

The first few numbers are:

**{1, 3, 6, 2, 7, 13, 20, …}.**

**Recamán sequence #2** (entry A8336 in the OEIS) is defined by:

- a
_{n+1}= a_{n}/ n if n divides a_{n}

The first few numbers are:

**{1, 1, 2, 6, 24, 120, …}.**

## Intuitive way to Create the Terms of the Recamán Sequence

Neil Sloane of ATT Labs (creator of OEIS) shed a little light on how the sequence is created in a 2008 Math Factor podcast [2].

Step 1: Write down the numbers

1 2 3 4 5 6 7 8 9

with gaps (we’re going to write the terms in those gaps).

Step 2: Write down the first term, which is 1, between the 1 and the 2:

1 (1) 2 3 4 5 6 7 8 9.

Step 2: Calculate the second term.

Ask yourself the question: Can we subtract 2 (the second number in our list) from 1 (the first term)?. We can’t (negative numbers or zero are not allowed), so we have to add 2 instead. 2 + 1 = 3, so:

1 (1) 2 (3) 3 4 5 6 7 8 9.

Step 2: Calculate the third term in the same way.

Ask yourself the question: Can we subtract 3 (the third number in our list from 3 (the second term). We can’t, so we have to add: 3 + 3 = 6, so:

1 (1) 2 (3) 3 (6) 4 5 6 7 8 9.

Continue in this way, obeying the basic rules:

**Numbers can’t be zero, negative, or something already in the sequence.**

If any of these conditions appear, just add instead of subtract. Note that if you add and get a number that has appeared before, that’s OK. It’s only not allowed if you subtract.

## Deciphering Recaman’s sequence

Although 10^{230} terms of Recaman’s sequence have been computed, but it still remains a mystery [2] despite the OEIS’s calculation of an enormous number of terms. No one knows if every number will eventually appear; Those dedicated to deciphering the sequence have dubbed it “How to Recamán’s life” [3].

As of the January 2018, Prime Curious! reported that the smallest missing prime in the Sequence is 966727 (as of January 2018).

## Random Sequence: References

[1] Ding, C. (2012). Sequences and Their Applications. Springer London.

[2] Math Factor Podcast. Retrieved April 9, 2021 from: http://mathfactor.uark.edu/2008/05/dw-the-online-encyclopedia-of-integer-sequences/

[2] Myers, J. et al. (2020). Three Cousins of Recaman’s Sequence. Retrieved April 9, 2021 from: https://ui.adsabs.harvard.edu/abs/2020arXiv200414000M/abstract

[3] Roberts, S. (2015). Genius At Play: The Curious Mind of John Horton Conway. Bloomsbury Publishing.

## Sequence and Series: What is a Series?

A series is what you get when you add up the terms in a sequence; when you add these terms, you’re performing what’s more formally called a summation (which is just a fancy way of saying “add them all up”).

## Finite Series

A **finite series** is a sum of a set amount of terms; A series of numbers (e.g. 1 + 2 + 3) is obtained from a sequence of numbers (e.g. 1, 2, 3); Finite series always have a first term and a last term. Plus, you can *always* find a solution for the **sum of a finite series.** For example, you can add up a given series of numbers (like 1 + 2 + 3 + 4) and find the answer (10).

A finite series and infinite series only differ from each other in terms of length. You can think of an infinite series as a “… huge or enormously long series” (Lazerowitz & Ambrose, 2016).

More formally, a finite series has the form (Dragomir & Sofo, 2008):

Where:

- Σ means to “sum up” (called sigma notation),
- a
_{i}; i = 1, 2, …, n) is a sequence of numbers.

## Finite Series Example: Finite Arithmetic Series

As a slightly more complicated example, the sum of the numbers 1 through 1000 is a finite sum: 500500:

This particular series is an example of an arithmetic series, which are defined by a common difference between each term (in this example, the difference is 1).

## Finite Series Formulas

Some of the more common you’ll come across (Sathaye, 2020):

Name | |

Arithmetic Series | |

Geometric Series | |

Telescoping Series |

## What is the Dirichlet Series?

The **Dirichlet series** has the form [1]:

Where:

*s*is a real-valued [2] or complex variable (Dirichlet proposed that*s*should be real or complex, but Riemann later on stressed the importance of*s*as a complex variable) [3].*f*(*n*) is a number theoretic function.

The notation *s* is a long standing tradition (going back to Dirichlet) and is written as *s* = σ + *it*, where *s* is the real part and *t* is the imaginary part. When σ > 1, the series converges absolutely and serves as a generating function for *f*(*n*).

There are hundreds of thousands of infinite and finite Dirichlet series of all kinds [2]. Two of the most well known are the Riemann zeta function, defined as the Dirichlet series associated with the constant function 1, and Dirichlet L-functions [3].

## Use of the Dirichlet Series

The series is often seen as a component for many generating functions for arithmetic functions. The method of generating functions is a powerful way of dealing with arithmetic functions. One type of generating functions is a type of power series:

f(0) + f(1)x + f(2)x^{2}…,.

The other type is the Dirichlet series:

The basic idea is that you relate the function *f* to the series F [3].

The Dirichlet series is a major component in proofs of the prime number theorem. For example, Hadamard and de la Vallee Poussin developed the first proof of the theorem by using the series to investigate the behavior of partial sums of arithmetic functions; Most proofs of the theorem relate the partial sums

to a complex integral involving the series