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, showed in red, is surrounded by a triangle of new points, in blue.
The first few centered triangular numbers are
1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971
Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centred triangular number has a remainder of 1 when divided by three and the quotient (if positive) is the previous regular triangular number.
Adding up the first n centered triangular numbers gives the magic constant for an n by n magic square (so long as n > 2).
