mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Mian–Chowla sequence is an

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

defined

recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

in the following way. The sequence starts with :

a_1 = 1.

Then for

n>1

a_n

is the smallest integer such that every pairwise sum :

a_i + a_j

is distinct, for all

i

and

j

less than or equal to

n

Properties

Initially, with

a_1

, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence,

a_2

, is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then,

a_3

can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that

a_3 = 4

, with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins : 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123,

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182 Year 182 ( CLXXXII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sura and Rufus (or, less frequently, year 935 ''Ab urbe condita'') ...

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, 252, 290, 361, 401, 475, ... .

Similar sequences

If we define

a_1 = 0

, the resulting sequence is the same except each term is one less (that is, 0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, ... ).

History

The sequence was invented by Abdul Majid Mian and

Sarvadaman Chowla Sarvadaman D. S. Chowla (22 October 1907 – 10 December 1995) was an Indian American mathematician, specializing in number theory. Early life He was born in London, since his father, Gopal Chowla, a professor of mathematics in Lahore, was then ...

References

* S. R. Finch, ''Mathematical Constants'', Cambridge (2003): Section 2.20.2 * R. K. Guy ''Unsolved Problems in Number Theory'', New York: Springer (2003) {{DEFAULTSORT:Mian-Chowla sequence Integer sequences