Obviously, there are an infinite number of possible sequences, and thousands have been studied. Some sequences contain deep insights into number theory. A sequence could be either infinite or finite. For simplicity, I will only consider sequences where the entries are real numbers (or infinity), although, you could have a sequence where the entries are any kind of mathematical object whatsoever.
A sequence has entries
a_1, a_2, a_3, a_4,…
One way to determine the entries, is to define a function f(n), and then evaluate it for positive integer values of n.
a_n = f(n)
f(1), f(2), f(3), f(4), …
I. Sequences From a Function
1. Constant function f(n) = c
c, c, c, c, …
2. Integers
f(n) = n
1, 2, 3, 4,…
3. Even Numbers
f(n) = 2n
2, 4, 6, 8,…
4. Odd Numbers
f(n) = 2n — 1
1, 3, 5, 7,…
5. Counting by c’s
f(n) = cn
c, 2c, 3c, 4c,…
So, for example, “counting by 10’s” would be 10, 20, 30, 40,…
6. Perfect Squares
f(n) = n²
1, 4, 9, 16, 25,…
7. Powers of 10
f(n) = 10^n
10, 100, 1000, 10000,…
8. Alternating Sequence
f(n) = (-1)^n
-1, 1, -1, 1,…
II. Sequences where Entries are Derived from Previous Entries
9. Doubling Sequence
a_(n+1) = 2a_n
1, 2, 4, 8, …
10. Halving Sequence
a_(n+1) = a_n/2
1, 1/2, 1/4, 1/8,…
11. Fibonacci Sequence = each term is the sum of the previous two terms
a_(n+2) = a_n + a_(n+1)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,…
If a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates the Golden Ratio, φ = a/b = (a+b)/a = 1.618033988…. The Fibonacci Sequence shows up repeatedly in botany. The number of petals of flowers are almost always Fibonacci numbers.
III. Sequences Related to Primes
12. Primes = positive integers greater than 1 that has no positive integer divisors other than 1 and itself
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…
Historically, “1” was considered prime, which meant that historically the primes coincided with the integers for the first three terms.
13. Composites = positive integers that are not prime
4, 6, 8, 9, 10, 12, 14,…
14. Prime Counting Function
pi(x) = number of primes less than x
0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6,…
15. Prime Sums
sigma(x) = sum of all primes less than x
2, 5, 10, 17, 28, 41, 58, 77,…
16. Twin Primes = primes where the nearest prime is 2 away
3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43,…
or making it explicit what the pairs are
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43),…
17. Cousin Primes = primes where the nearest prime is 4 away
18. Sexy Primes = primes where the nearest prime is 6 away
19. Prime Gaps = differences between consecutive primes
1, 2, 2, 4, 2, 4, 2, 4, 6, 2,…
20. Mersenne Primes = a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n
3, 7, 31, 127,…
21. Fermat Numbers = numbers of the form 2²^n + 1
3, 5, 17, 257, 65537, 4294967297, 18446744073709551617,…
22. Fermat Primes = Fermat number that is prime
3, 5, 17, 257, 65537
It is believed that these are the only five terms in the sequence.
23. Pseudoprimes = A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers. There are actually different types of pseudoprimes based on what tests it passes or what qualities it shares with prime numbers. Some examples of pseudoprimes are Fermat pseudoprimes, Poulet numbers, Euler-Jacobi pseudoprimes, strong pseudoprimes, and Carmichael numbers.
24. Perfect Numbers = positive integers that are equal to the sum of their proper divisors
6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128,…
25. Amicable Numbers = one of a pair of positive integers where each is equal to the sum of their proper divisors of the other
220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595,…
or making it explicit what the pairs are
(220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595),…
26. Fibonacci Primes = Fibonacci number that is prime
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073,…
IV. Other Sequences
27. Hiccup Sequence = integers with an extra “1”
1, 1, 2, 3, 4,…
28. Asymptotic to Pi
3, 3.1, 3.14, 3.145, 3.14159,…
29. Look Up and Say
1, 11, 21, 1211, 111221, 312211, 13112221,…
You read the numbers out loud. When you see “1”, you say there is “one one”. When you see “11”, you say there are “two ones”, etc.
30. Palindrome Numbers
1, 2, 3, 4,… other examples include 11, 22, 33, 101, 121, 424, 1001, 246642, 332233, 100040001, etc.
31. Number of Regular Examples of Polytopes in Each Dimension
1, 1, infinity, 5, 6, 3, 3, 3,…
The only polytope in zero dimensions is the point. The only polytope in one dimension is the line segment. Both could be considered regular. In two dimensions, there are an infinite number of regular polygons. In three dimensions, there are five regular polyhedra, which are the five platonic solids, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In the four dimensions, you have 4D analogs of each of these, plus an extra one called the 24-cell. In five or more dimensions, you only have analogs of the tetrahedron, cube, and octahedron.
32. Dimensions of Normed Division Algebras
1, 2, 4, 8
There are only four normed division algebras. The real numbers are one dimensional. The complex numbers are two dimensional. The quaternions are four dimensional. The octonions are eight dimensional.
This example shows that you can not guess what the next term is in a sequence, or even if there is a next term, from simply looking at the first few terms. If you looked at the above sequence, you might incorrectly guess that the next term was 16, when, in reality, the correct answer is that there is no next term because it is a finite sequence with only four terms.
In Elementary school, a student might be asked to write down the next few terms in the sequence
5, 10, 15, 20,…
The teacher expects you to write
5, 10, 15, 20, 25, 30, 35, 40,…
If the student wrote
5, 10, 15, 20, 1, 2, 3, 4,…
the teacher would mark it wrong. However, that is a completely valid answer. That is just as much an acceptable sequence as the one the teacher was looking for. The teacher might say, “Assume the pattern holds”. Well, from just looking at the first four terms, we do not know what the pattern is. Maybe the pattern is this.
5, 10, 15, 20, 1, 2, 3, 4, 5, 10, 15, 20, 1, 2, 3, 4, 5, 10, 15, 20, 1, 2, 3, 4,…
The truth is, in order to calculate the terms in a sequence, you must be given the definition of the sequence. The first few terms alone give zero indication of what the next term will be.
Often on the Internet, you see so-called puzzles where you are supposed to guess what comes next in the sequence. Of course, from the point of view of mathematics, any answer is equally valid, whether or not it is what the writer had in mind. It not a test of mathematics. It is a test in psychology, trying to guess what the writer probably meant.
There are sequences where we do not know whether it has zero terms or an infinite number of terms. Consider the following sequence.
33. Exceptions to the Riemann hypothesis
If the Riemann hypothesis is true, then there are zero terms in this sequence. If it is not true, then there are an infinite number of terms in this sequence. Right now, we do not know which it is.
34. Coefficients of Modular Function j
1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600,…
35. Degrees of irreducible representations of Monster group M
1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999, 125510727015275, 190292345709543,…
The above two sequences are related to Monstrous Moonshine. In November 1978, J. McKay noticed that the [q bar] -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the Monster Group (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of j(tau) can be expressed as linear combinations of these dimensions with small coefficients.
This was a very surprising connection between two seemingly unrelated branches of mathematics. It turned out to have deep connections to string theory. The name “Moonshine” was a reference to the phrase “talking moonshine” meaning saying something untrue, although in this case, it did turn out to be true.
If someone gave you the first few terms of a sequence, and asked you to guess the next one, you could choose any number at random, and then write down a polynomial formula that produces that sequence, including the last term that you made up.
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