centeredrectangular

The centered rectangular number is a figurate number that represents a rectangle with a central point. It is formed by taking a rectangle with dimensions $2n \times (2n+1)$ and adding a single central point. These numbers can be expressed by the formula $C_n = 2n(2n+1) + 1$. The sequence begins with $n=0$.

The first few centered rectangular numbers are 1, 9, 25, 49, 81, 121, and so on. This sequence can also be described as the sequence of odd squares. The formula $C_n = (2n+1)^2$ yields the same sequence. For example, when $n=0$, $C_0 = (2(0)+1)^2 = 1^2 = 1$. When $n=1$, $C_1 = (2(1)+1)^2 = 3^2 = 9$. When $n=2$, $C_2 = (2(2)+1)^2 = 5^2 = 25$.

The difference between consecutive centered rectangular numbers is given by $C_n - C_{n-1} = (2n+1)^2 - (2(n-1)+1)^2 = (2n+1)^2 - (2n-1)^2$. This difference is $(4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n$. Thus, the differences between successive terms are multiples of 8: 8, 16, 24, 32, etc. The $n$-th centered rectangular number can also be obtained by summing the first $n+1$ odd numbers, starting with 1.