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:
“Maturity involves being honest and true to oneself, making decisions based on a conscious internal process, assuming responsibility for one’s decisions, having healthy relationships with others and developing one’s own true gifts. It involves thinking about one’s environment and deciding what one will and won’t accept.”
—Mary Pipher (20th century)
“The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
—Claude Lévi-Strauss (b. 1908)