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The triangular number $T_n$ is the total number of objects arranged in a triangle of $n$ rows, i.e. $T_n:=\dfrac{n(n+1)}{2}$. Then, one might ask for the triangular root of a number $x$, as the positive number $n$ (not necessarily an integer) such that $T_n=x$. Clearly, $n=\dfrac{\sqrt{8x+1}-1}{2}$, but:

Question(s): Is there any reasonably standard notation for this root? If needed, would perhaps $\sqrt[^\Delta]{x}$ be appropriate?

It seems like a very simple question, but I couldn't seem to find any. Wikipedia suggests none as far as I can see, and following Wikipedia's reference to Euler's Elements of Algebra I could not seem to find where they where even defined, but I may have missed it. Also, I am very unfamiliar with number theory, and most searches seem to result in material related to cube or even higher $n$:th roots.

Christopher.L
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    Does this function need a special notation? – Yuriy S Apr 28 '21 at 13:16
  • I suspect that $T^{-1}_n$ would be preferred notation as it connects clearly with the triangular number $T_n$ and that's actually what you've found. Square roots don't have a simple closed form formula. – postmortes Apr 28 '21 at 13:17
  • @YuriyS Perhaps not. I just found myself in the need of one when working on my thesis, as I refer to it occasionally. Hence I had to write something like "Let $t(x)$ be the triangular root of $x$..." and so on. So, it's not that I can not live without it, it's just that I thought that there might be one. But, perhaps it's not that useful after all, and hence why I couldn't find any. – Christopher.L Apr 28 '21 at 13:21
  • @postmortes : Yes, I guess that's sufficient of course; probably why there was no such notation to be found as well. I guess then one would write $T^{-1}_x$ in my case. – Christopher.L Apr 28 '21 at 13:26
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    I just found a very non-authoritative source, which at least shows that your $\sqrt[^\Delta]{x}$ idea isn't that far-fetched. Personally, I like it. – user3733558 Apr 28 '21 at 13:46
  • @user3733558 Well, that's what felt most natural at first. But, as is pointed out, since I already use $T_n$, I guess $T^{-1}_x$ would be most appropriate. So, I also like it, possibly because it seems a bit unique and intuitive out of context, but I will of course use what will make most sense in context, which is $T^{-1}_x$. – Christopher.L Apr 28 '21 at 14:06

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The OEIS uses $\operatorname{trinv}(x)$ for $\left\lfloor\frac{\sqrt{8x+1}+1}2\right\rfloor$, which differs from your function by $1$; it appears 43 times as of this post's writing.

Parcly Taxel
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