Centered triangular number

A centered (or centred) 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

{\frac {3n^{2}+3n+2}{2}}.

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:

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, … (sequence A005448 in the OEIS).

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 (if positive) 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.

Gnomon

The gnomon of the n'th centered triangular number is : ${\frac {3(n+1)^{2}+3(n+1)+2}{2}}-{\frac {3n^{2}+3n+2}{2}}={\frac {3n^{2}+9n+8-3n^{2}-3n-2}{2}}={\frac {6n+6}{2}}=3n+3.$

References

Lancelot Hogben: Mathematics for the Million.(1936), republished by W. W. Norton & Company (September 1993), ISBN 978-0-393-31071-9
Weisstein, Eric W. "Centered Triangular Number". MathWorld.