Triangular numbers
The $n$th triangular number is
$T_n =1+2+3+ ··· + n = \dfrac{1}{2}n(n + 1)$.
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
$P_n = \dfrac{1}{2}n(3n − 1)$.
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
$P_{k,n} = \dfrac{1}{2}n((k − 2)n − (k − 4))$.
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