Centered figurate number that represents a triangle with a dot in the center

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).

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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.

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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} < 10^{9}) 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

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