Figurate numbers: Triangular, Pentagonal Polygonal numbers

Triangular numbers

The $n$th triangular number is

$T_n=1+2+3+···+n={\displaystyle \frac{1}{2}}n(n+1)$.

The first few of these are 

$1,3,6,10,15,21,28,36,45,55,....$

Pentagonal numbers

The pentagonal numbers are the sums of the arithmetic progression

$1+4+7+···+(3n-2)+···$.

The nth pentagonal number is

$P_n={\displaystyle \frac{1}{2}}n(3n-1)$.

The polygonal numbers $P_{n,k}$

More generally, for a fixed $k$, the $k$-gonal numbers are the sums of the arithmetic progression 

$1+(k-1)+(2k-3)+···$. 

The nth $k$-gonal number is 

$P_{k,n}={\displaystyle \frac{1}{2}}n((k-2)n-(k-4))$.