A series of numbers generated from the expression x(x + 1) - ie. the nth 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 nth 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 xn = 1 + 4n, 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

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