Triangular numbers

From specialfunctionswiki
Jump to: navigation, search

The triangular numbers $T(n)$ are defined for $n=1,2,3,\ldots$ by the formula $$T(n)=\displaystyle\sum_{k=1}^n k.$$ They represent the number of ways to draw an equilateral triangle as in the image below.

Properties[edit]

T(n)=n(n+1)/2
T(n+1)=T(n)+n+1
n^2=T(n)+T(n-1)
T(n)^2=T(T(n))+T(T(n)-1)
T(n+1)^2-T(n)^2=(n+1)^3

References[edit]

Polygonal numbers