A

series of

numbers generated from the

expression
`x`(

`x` + 1) -

*ie.* the

`n`th
term is

`n`(

`n` + 1). The first few terms are:

`0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162`

The **pronic** or
**heteromecic** numbers have many interesting properties.

Each term in the series is exactly twice the corresponding term in the series of triangular numbers as shown below:

o
o oo
o oo ooo
o oo ooo oooo
o oo ooo oooo ooooo
triangular: 1 3 6 10 15
ooooo
oooo oooo
ooo ooo o ooo
oo oo o oo oo oo
o o o o oo o ooo o
o oo ooo oooo
o oo ooo oooo ooooo
pronic: 2 6 12 20 30

Because the

`n`th term in the series of triangular numbers is also the sum of all integers up to and including

`n` we can say that the sum of all the integers from 1 to

`n` can be calculated by the formula

`n`(

`n` + 1)/2.

For example the sum of all the integers from 1 to 10, 1 + 2 + 3.. + 10 is 10(10 + 1)/2, or 110/2, giving 55.

Pronic numbers can crop up in some odd places. For example, take the series generated by `x`_{n} = 1 + 4`n`, and then take only the pronic terms (the second, 6th, 12th, *etc.* terms) and you will obtain the squares of the odd integers (1, 9, 25 ...) but take the terms midway between these terms, and you get the squares of the even integers, **plus one!**, (17, 37, 65...). In fact the
numbers "midway between the pronic numbers" are just the squares of the natural numbers, (1, 4, 9, 16 ...)

You can look up the pronic numbers, and many other integer
sequences at the **On-Line Encyclopaedia of Integer Sequences**
at: http://www.research.att.com/~njas/sequences/Seis.html