mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 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, 148, 182, 204,

252 Year 252 ( CCLII) was a leap year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Trebonianus and Volusianus (or, less frequently, year 1005 ''Ab urbe condita''). The denomination 252 for t ...

, 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.

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