Editing Centered triangular number (section)

==Properties==
*The [[Gnomon (figure)|gnomon]] of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is:

::<math>C_{3,n+1} - C_{3,n} = 3(n+1).</math>

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

::<math>C_{3,n} = 1 + 3 \frac{n(n+1)}{2} = \frac{3n^2 + 3n + 2}{2}.</math>

*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 number]]s.

*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:

:<math>C_{3,n} = \frac{3C_{4,n} + 1}{4},</math>

where

:<math>C_{4,n} = n^{2} + (n+1)^{2}.</math>