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The Triangular Number is calculated by summing up the numbers of the function y=x at an integer interval. So its

tri(0)=0
tri(1)=1
tri(2)=3
tri(3)=6

https://en.wikipedia.org/wiki/Triangular_number

So whats it called if instead of the function y=x we have another function, (ie y=x^2, y=log(x), y=ax^2+bx+c)?

user2741831
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  • The sum of the first squares are called square pyramidal numbers. – angryavian Jan 04 '21 at 15:06
  • tri(0) by your and the usual definition should have been zero, not one. Recall the basic properties of the empty sum and summations in general. – JMoravitz Jan 04 '21 at 15:10
  • yep, that was a typo – user2741831 Jan 04 '21 at 15:11
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    As for the completely general name for what you are after... there isn't one beyond "series" or "summation"... what you are describing are already so incredibly general they get a generic name. If you are looking very specifically at sums of the form $y=x^k$ (using your notation) see Faulhaber's formula. – JMoravitz Jan 04 '21 at 15:13
  • thanks. Altough Its pretty incredible to me that a problem this simple would have such a complicated solution. – user2741831 Jan 04 '21 at 15:19
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    Do not confuse "simple to ask" with "simple to solve" or with "simple to state the answer"... these often have nothing to do with one another. As for sums of squares, sums of cubes, and so on... they are only as "complicated" as you allow them to be. On the contrary, I find the general solution quite elegant. – JMoravitz Jan 04 '21 at 15:23
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    If "$\operatorname{tri}f$" indicates your triangular number idea, then basically you're defining $$\operatorname{tri}_f(n) = f(0)+f(1)+f(2)+\cdots+f(n) = \sum{k=0}^n f(k)$$ As @JMoravitz notes, we call such a thing a "series" or "summation". Whether there's a nice formula for the result (what's we call a "closed form") depends entirely on the nature of $f$. (It's safe to say that "most" series don't have an elementary closed form.) A site search for the tag sequences-and-series shows how complicated these can get. – Blue Jan 04 '21 at 15:24
  • Since Faulhabers Formula includes a sum over 2 and p, is it safe to assume this won't work with fractional exponents such as 1.1? – user2741831 Jan 04 '21 at 15:26
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    Yes, Faulhaber's formula was specifically for the case of constant positive integer exponents. If you are interested in $\sum\limits_{x=1}^n x^p$ for non-integer $p$, see this post and also see the related Riemann Zeta Function if we were to let the upper limit of the sum go to infinity. – JMoravitz Jan 04 '21 at 15:32
  • thanks, I'll look into it – user2741831 Jan 04 '21 at 15:42

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