Suppose I have an integer sequence $\{a_k\}$, and suppose I know a closed form in terms of $n$ for this sum:

$$\displaystyle\sum\limits_{k=1}^{n} a_k$$

Given this, is it always (or ever) possible to find a closed form in terms of $n$ for

$$\displaystyle\sum\limits_{k=1}^{n} a_k^2$$

or to say anything about this sum?

If not, what special cases allow us to extend the closed form to the sum of squares in some way?

  • $\begingroup$ This was inspired by this question regarding a series related to the hypertriangular function of $n$. $\endgroup$ – Zubin Mukerjee Dec 21 '14 at 12:38
  • $\begingroup$ The simplest case consists of the whole numbers and squares of whole numbers. You have $\Sigma n = n(n+1)/2$, $\Sigma n^2 = n(n+1)(2n+1)/6$. Is there a way to find the (latter) sum of squares using only the formula for $\Sigma n$? $\endgroup$ – user_of_math Dec 21 '14 at 17:27

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