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 words relationships with and/or numbers:
“She has problems with separation; he has trouble with unity—problems that make themselves felt in our relationships with our children just as they do in our relations with each other. She pulls for connection; he pushes for separateness. She tends to feel shut out; he tends to feel overwhelmed and intruded upon. It’s one of the reasons why she turns so eagerly to children—especially when they’re very young.”
—Lillian Breslow Rubin (20th century)
“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)