To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # Pronic number

Demonstration, with Cuisenaire rods, of pronic numbers n=1, n=2, and n=3 (2, 6, and 12).

A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n(n + 1). The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the "rectangular number" name has also been applied to the composite numbers.

The first few pronic numbers are:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 … (sequence A002378 in the OEIS).

If n is a pronic number, then the following is true:

$\lfloor {\sqrt {n}}\rfloor \cdot \lceil {\sqrt {n}}\rceil =n$ • 1/1
Views:
303
• ✪ MathBabbler: Pronic Numbers (and Remembering Billy Powell)

## As figurate numbers

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics, and their discovery has been attributed much earlier to the Pythagoreans. As a kind of figurate number, the pronic numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:

The nth pronic number is twice the nth triangular number and n more than the nth square number, as given by the alternative formula n2 + n for pronic numbers. The nth pronic number is also the difference between the odd square (2n + 1)2 and the (n+1)st centered hexagonal number.

## Sum of pronic numbers

The sum of the reciprocals of the pronic numbers (excluding 0) is a telescoping series that sums to 1:

$1={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{12}}\cdots =\sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}.$ The partial sum of the first n terms in this series is

$\sum _{i=1}^{n}{\frac {1}{i(i+1)}}={\frac {n}{n+1}}.$ The partial sum of the first n pronic numbers is twice the value of the nth tetrahedral number:

$\sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}={\frac {n^{\overline {3}}}{3}}.$ $(10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,$ .