Centered octahedral number
When we talk about centered octahedral numbers, it means we are counting the number of objects that can be arranged in a specific way. Imagine you have a bunch of balls and you want to arrange them in a special way. You start by putting one ball in the center and then surround it with 6 other balls. These 6 balls will touch the center ball and each other, forming something that looks like a octahedron (an octahedron is a shape that has 8 faces and looks like two pyramids stuck on top of each other). Now, if you continue adding more layers to this arrangement, you will eventually end up with a bigger and bigger octahedron made up of more and more balls.
The number of balls that you need to make a specific sized octahedron is called a centered octahedral number. For example, if you want to make a small octahedron with 7 balls, you will need to have 1 ball in the center and 6 balls around it. If you want to make a bigger octahedron with 15 balls, you will need to start with 1 ball in the center and then surround it with 6 balls, then surround those with 12 balls, and finally surround those with 18 balls. The total number of balls used to make this big octahedron is 1+6+12+18 = 37, so we say that 37 is a centered octahedral number.
In summary, centered octahedral numbers are a way of counting the number of objects needed to create a specific arrangement of objects that look like a octahedron. They start with a center object and then add layers of objects around it in a specific pattern to make bigger and bigger octahedrons.
The number of balls that you need to make a specific sized octahedron is called a centered octahedral number. For example, if you want to make a small octahedron with 7 balls, you will need to have 1 ball in the center and 6 balls around it. If you want to make a bigger octahedron with 15 balls, you will need to start with 1 ball in the center and then surround it with 6 balls, then surround those with 12 balls, and finally surround those with 18 balls. The total number of balls used to make this big octahedron is 1+6+12+18 = 37, so we say that 37 is a centered octahedral number.
In summary, centered octahedral numbers are a way of counting the number of objects needed to create a specific arrangement of objects that look like a octahedron. They start with a center object and then add layers of objects around it in a specific pattern to make bigger and bigger octahedrons.