Centered Triangular Number

A centered triangular number is a centered figurate number formed by an equilateral triangle with one dot in the middle and further dots ringing that center in successive triangular layers. The nth centered triangular number is given by the formula three times n squared, plus three times n, plus two, all divided by two, which is the same as one plus three times n times n plus one, divided by two. The sequence begins one, four, ten, nineteen, thirty-one, forty-six, sixty-four, eighty-five, one hundred nine, one hundred thirty-six, and every term in it leaves a remainder of one when divided by three. From the third term on, each centered triangular number is also the sum of three consecutive ordinary triangular numbers.