This time not stupid math

**SKG_Scintill** · 03-20-2013, 06:27 AM

But it is a problem.
I'll leave out the context, because it's a bit extensive.

These are the sequences I'm using:
Triangular numbers: $\sum\limits_{k=0}^\infty k^1=\{1,3,6,10,15,\cdots\}$
Square pyramidal numbers: $\sum\limits_{k=0}^\infty k^2=\{1,5,14,30,55,\cdots\}$
(Does it have a name?) numbers: $\sum\limits_{k=0}^\infty k^3=\{1,9,36,100,225,\cdots\}$
And so on and so on, each time increasing the power.

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These are (respectively) the formulas for the first two sequences:
Triangular numbers: $\dfrac{n(n+1)}{2}$
Square pyramidal numbers: $\dfrac{n(n+1)(2n+1)}{6}$

Now what I want to know:
1. Does the third sequence have a name? :P
2. What is the formula for the third sequence?
3. How do you translate the formula of one sequence into the formula of the next sequence?

I've tried a trial-and-error method, but that's prone to fail anyway. Does anyone know how to do it mathematically?

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P.S.: why are \cdots acting up? Solved it, not sure how.

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