On Generalization of Centred Odd Cube of Length 2r+1
$\sum_{i=1}^{2r+1}w_i^{3} = \left(\sum_{i=1}^{2r+1}w_i\right) \left(m^{2}+r(r+1)d^{2}\right)$
Abstract
The study of integer $I$ for which $I$ is a sum of cubes is a subject of study in the areas of Diophantine equations, perhaps this is due to the fact that Diophantine equations have direct applications in coding theory. Let $r, w, d$ and $m$ be positive integers and suppose that $d$ is a common difference between any two integers in the sequence $w_{i+1}$ and $w_i$, where $m=w_{r+1}$ denotes the middle term of the sequence. The current paper is therefore set to develop and introduce the general formula for the sum of an odd number of centred odd cubes. In particular, the formula $\sum_{i=1}^{2r+1}w_i^{3}=\left(\sum_{i=1}^{2r+1}w_i\right)\left(m^{2}+r(r+1)d^{2}\right)$ is introduced. The methodology involves expressing the arithmetic sequence in its standard form, expanding the cube terms, and matching coefficients. Several examples are presented to verify the identity and illustrate its computational efficiency.
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