# Pronic number

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 term "rectangular number" 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:

$\lfloor {\sqrt {n}}\rfloor \cdot \lceil {\sqrt {n}}\rceil =n$ ## 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:                                        1×2 2×3 3×4 4×5

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:

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

$(10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,$ .