Centered triangular number

A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.

This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to $n$ .

The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue).

Properties

The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-th triangular layer, is:

C_{3, n + 1} - C_{3, n} = 3 (n + 1) .

The n-th centered triangular number, corresponding to n layers plus the center, is given by the formula:

C_{3, n} = 1 + 3 \frac{n (n + 1)}{2} = \frac{3 n^{2} + 3 n + 2}{2} .

Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number.

Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers.

For n > 2, the sum of the first n centered triangular numbers is the magic constant for an n by n normal magic square.
It is the difference of two tetrahedral numbers given by the formula: $\begin{matrix} C_{3, n} = T e_{n} - T e_{n - 3}, n > 3 \end{matrix}$

Relationship with centered square numbers

The centered triangular numbers can be expressed in terms of the centered square numbers:

C_{3, n} = \frac{3 C_{4, n} + 1}{4},

where

C_{4, n} = n^{2} + (n + 1)^{2} .

Lists of centered triangular numbers

The first centered triangular numbers (C_3,n < 3000) 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).

The first simultaneously triangular and centered triangular numbers (C_3,n = T_N < 10⁹) are:

1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … (sequence A128862 in the OEIS).

The generating function

If the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all $| x | < 1$ , in which case it can be expressed as the meromorphic generating function

1 + 4 x + 10 x^{2} + 19 x^{3} + 31 x^{4} + ... = \frac{1 - x^{3}}{(1 - x)^{4}} = \frac{x^{2} + x + 1}{(1 - x)^{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.