Centered triangular number
A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for n is given by the formula
The following image shows the building of the centered triangular numbers using the associated figures: at each step the previous figure, shown in red, is surrounded by a triangle of new points, in blue.
The first few centered triangular numbers are:
Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centered triangular number has a remainder of 1 when divided by three and the quotient is the previous regular triangular number.
The sum of the first n centered triangular numbers is the magic constant for an n by n normal magic square for n > 2.
