Centered cube number

A centered cube number is a number that is the sum of two consecutive cube numbers . For example, is a centered cube number. The first cube numbers centered are ${\displaystyle 35=8+27=2^{3}+3^{3}}$

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, ... (sequence A005898 in OEIS )

The centered cube numbers are the spatial extension of the centered square numbers into the third dimension .

calculation

The -th centered cube number is calculated using the formula ${\displaystyle n}$ ${\displaystyle ZK_{n}}$

${\displaystyle ZK_{n}=n^{3}+(n-1)^{3}=2n^{3}-3n^{2}+3n-1}$

Relationships to other figured numbers

The -th centered cube number is the sum of the first centered square numbers . ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle ZK_{n}=\sum _{k=1}^{n}ZQ_{k}=ZQ_{1}+...+ZQ_{n}}$

properties

• All centered cube numbers are odd.
• The following applies, where the -th quadratic pyramidal number is: ${\displaystyle Pyr_{n}}$ ${\displaystyle n}$

${\displaystyle ZK_{n}=Pyr_{n}+4\cdot Pyr_{n-1}+Pyr_{n-2}.}$

• The sum of the reciprocal values ​​of the centered cube numbers, so is convergent. ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{ZK_{k}}}}$
• The form of centered cubic numbers occurs naturally in the structure of atoms.

Generating function

The function

${\displaystyle {\frac {x(x^{3}+5x^{2}+5x+1)}{(x-1)^{4}}}=x+9x^{2}+35x^{3}+91x^{4}+\ldots }$

contains in its series expansion on the left side of the equation the sequence of the centered cubic numbers. It is therefore called the generating function of the sequence of centered cube numbers.

Web links

• Page by Jutta Gut about three-dimensional figured numbers
• Eric W. Weisstein : Centered Cubic Number . In: MathWorld (English).