Relationships With Other Figurate Numbers
The nth centered square number is given by the formula
In other words, a centered square number is the sum of two consecutive square numbers. The following pattern demonstrates this formula:
The formula can also be expressed as
that is, n th centered square number is half of n th odd square number plus one, as illustrated below:
Like all centered polygonal numbers, centered square numbers can also be expressed in terms of triangular numbers:
where
is the nth triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:
The difference between two consecutive octahedral numbers is a centered square number (Conway and Guy, p.50).
Read more about this topic: Centered Square Number
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