Centered triangular number
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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 .
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).
Relationship with centered square numbers
- The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-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 (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.
The centered triangular numbers can be expressed in terms of the centered square numbers:
where
