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).

construction

File:Centered triangular numbers hex grid.svg

The first eight centered triangular numbers on a hex grid

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:

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

• 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.

Relationship with centered square numbers

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

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

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

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

References

• Lancelot Hogben: Mathematics for the Million (1936), republished by W. W. Norton & Company (September 1993),