Centered triangular number


A centered '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.
The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle is surrounded by a triangular layer of new dots.

Properties

  • The gnomon of the n-th centered triangular number, corresponding to the -th triangular layer, is:
  • The n-th centered triangular number, corresponding to n layers plus the center, is given by the formula:
  • Each centered triangular number has a remainder of 1 when divided by 3, and the quotient 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:
where

Lists of centered triangular numbers

The first centered triangular numbers are:
The first simultaneously triangular and centered triangular numbers are:

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, in which case it can be expressed as the meromorphic generating function