The world of mathematical sequences and series is quite fascinating and absorbing. Such sequences are a great way of mathematical recreation. The sequences are also found in many fields like Physics, Chemistry and Computer Science apart from different branches of Mathematics. Only a few of the more famous mathematical sequences are mentioned here:
(1) Fibonacci Series: Probably the most famous of all Mathematical sequences; it goes like this—- 1,1,2,3,5,8,13,21,34,55,89…
At first glance one may wonder what makes this sequence of numbers so sacrosanct or important or famous. However a quick inspection shows that it begins with two1 s and continues to get succeeding terms by adding, each time, the last two numbers to get the next number (i.e., 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on).
By itself, this is not very remarkable. Yet there are no numbers in all of mathematics as all-pervading as the fabulous Fibonacci numbers. They pop up every now and then in nature, geometry, algebra, number theory, Permutations and combinations and many other branches of mathematics. More stunningly, they appear in nature abundantly; for example, the number of spirals of bracts on a pinecone is always a Fibonacci number, and, similarly, the number of spirals of bracts on a pineapple is also a Fibonacci number. The appearances in nature seem boundless. The Fibonacci numbers can be found in connection with the arrangement of branches on various species of trees, as well as in the number of ancestors at every generation of the male bee on its family tree. There is practically no end to where these numbers appear or be sighted.
Fibonacci numbers are very much connected to the famous ‘Golden Ratio’ or ‘Divine ratio’ whose value is equal to 1.618…
The larger the Fibonacci numbers, the closer their ratio of last two terms approaches the golden ratio. For example, the quotient of the relatively small pair of consecutive Fibonacci numbers:
Now, consider the quotient of the somewhat larger pair of consecutive
These increasingly larger quotients seem to surround, the actual value of the golden ratio. When we take much larger pairs of consecutive Fibonacci numbers, their quotients get us ever closer to the actual value of the golden ratio.
There are many properties of Fibonacci series, only a few are listed below:
i. The sum of any ten consecutive Fibonacci numbers is divisible by 11.
ii. Two consecutive Fibonacci numbers do not have any common factor, which means that they are Co-prime or relatively prime to each other.
iii. The Fibonacci numbers in the composite-number (i.e., non-prime) positions are also composite numbers.
iv. The sum of the first n Fibonacci numbers is equal to the Fibonacci number two further along the sequence minus 1.Mathematically , F1 + F2+F3……..+Fn = Fn+2 -1.
There are many counting problems in combinatorics whose solution is given by the Fibonacci Numbers.
(2) Figurate Numbers series like square, triangular, pentagonal, hexagonal no. series.
(a) Square Numbers Series: it is quite self explanatory: 1, 4,9,16,25,36,49…
Pictorially, the square numbers can be represented as below:
(b) Triangular number Series: A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangle number is the number of dots or balls in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n.
Pictorially, the triangular numbers can be represented as below:
The sequence of triangular numbers is: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ….
(c) Pentagonal number Series: A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon. A pentagonal is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176…
Pictorially, the Pentangular numbers can be can be represented as below:
(d) Hexagonal Numbers: Similarly, pictorially, the hexagonal numbers can be represented as below:
The formula for the nth hexagonal number:
The first few hexagonal numbers are:
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231…
(e) The Lazy Caterer’s Sequence: Formally also known as the central polygonal numbers, it describes the maximum number of pieces (or bounded/unbounded regions) of a circle (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but seven if they do not.
The maximum number p of pieces that can be created with a given number of cuts n, where n ≥ 0, is given by the formula
Using binomial coefficients, the formula can be expressed as
(3) Magic Square series: In recreational mathematics, a magic square of order ‘n’ is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2. The term “magic square” is also sometimes used to refer to any of various types of word square.
The constant sum in every row, column and diagonal is called the magic constant or magic sum, M. The magic constant of a normal magic square depends only on n and has the value
Thus the magic square series is like this: 15, 34, 65, 111, 175, 260…
(5) Catalan Number Series: In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. The nth Catalan number is given directly in terms of binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3 … are
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862…
There are many counting problems in combinatorics whose solution is given by the Catalan numbers.
(6) Look and say sequence: In mathematics, the look-and-say sequence is the sequence of integers as below:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …
To generate a term of the sequence from the previous term, just ‘look and say or read’ the digits of the previous member. For example:1 is read off as “one 1” or 11.11 is read off as “two 1s” or 21.21 is read off as “one 2, then one 1” or 1211.1211 is read off as “one 1, then one 2, then two 1s” or 111221.111221 is read off as “three 1s, then two 2s, then one 1” or 312211.( The look-and-say sequence was introduced and analyzed by John Conway in his paper “The Weird and Wonderful Chemistry of Audioactive Decay” published in Eureka 46, 5–18 in 1986.However it has great recreational value and it has appeared in several Management Entrance exams in past.)
Finally a few special series are mentioned below from other branches than Mathematics:
(a) 1, 6, 30, 138, 606… It is about susceptibility for the planar hexagonal lattice2 in Physics.
(b) 1, 1, 4, 8, 22, 51… It is about the numbers of alkyl derivatives of benzene with n=6,7,…,carbon atoms.
(c) 3, 7, 46, 4436, 134281216… in Electrical Engineering about Boolean functions of n variables.
(d) 0, 1, 3 , 5 , 9, 11, 14, 17, 25, 27 , . ..In Computer Science about the number of comparisons needed to sort n elements by list merging.
I hope that readers will find this article on famous sequences in mathematics both interesting and stimulating.