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
Famous quotes containing the word numbers:
“Individually, museums are fine institutions, dedicated to the high values of preservation, education and truth; collectively, their growth in numbers points to the imaginative death of this country.”
—Robert Hewison (b. 1943)