Sum of the first n terms of Central polygonal numbers (the Lazy Caterer's sequence) How do you find the sum of the first $n$ terms of the Central polygonal numbers (the Lazy Caterer's sequence): A000124 - OEIS $1, 2, 4, 7, 11, 16, 22, 29, 37, 46, \ldots$ I tried but could not solve. Please help!!!! I am from Brazil.

There are many ways of deriving the $i$-th term.

One easy way is the following:

Notice that if the $i$-th term is $t_i$ then

$t_2 - t_1 = 1$

$t_3 - t_2 = 2$

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$t_i - t_{i-1} = i-1$

Add all the above equations to obtain

$t_i - 1 = \frac{i(i-1)}{2}$

or, $t_i = \frac{1}{2}(i^2 - i + 2)$

Finally

$\displaystyle \sum_{i=1}^n t_i = \frac{1}{2} \left(\displaystyle \sum_{i=1}^n i^2 - \displaystyle \sum_{i=1}^n i + \displaystyle \sum_{i=1}^n 2 \right) = \frac{1}{2} \left[\frac{n(n+1)(2n+1)}{6} - \frac{n(n+1)}{2} + 2n\right] = \frac{n(n^2+5)}{6}$