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Here I am again, trying to organize my thought. Thank you Don for encouraging me yesterday. (From this thread Asking for suggestions about square numbers)

I'll try to explain as much as I can, but some parts I really don't know how to explain them...

When you look at the differences between each of the sq number, you find that the differences increase steadily with +2 to each. I call it layers until steady. (Just want some words to identify it).

layers of ^2

We can see that numbers of layers agree with number of the ^ (I don't know what is the word for the ^....). See the ^3 layers

layers of ^3

Same goes with ^4 and ^5 and above

layers of ^4

layers of ^5

Now let's take a look at a concrete picture of 3^2 and 4^2

3^2 and 4^2

If we want to find their differences, we pull 1 from each ball from 4^2 (to make each ball has 3), we need to pull only 3 balls because we're making them 3^2; and we have one of the 4^2 ball left.

Thus, 4^2 - 3^2 = [3x3](3, which remain in each ball after we pulled 1 out) + 3(pulled from three of the 4^2 balls) + 4 (that ball left from 4^2)

See more picture, please.

20^2 and 21^2

22^2

60^2 and 61^2

I'll continue in here (Continue) About square numbers because I my reputation is not enough to post more than 8 links.

PacharapanK
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  • Oh... we can't add photos in comments T_T what should I do – PacharapanK Oct 28 '19 at 18:40
  • If you could summarise your question in 1 sentence, what is it? (Also I haven't seen those circular and rectangular diagrams before) – helpme Oct 28 '19 at 18:48
  • Dear helpme, it's from this thread https://math.stackexchange.com/questions/3411923/asking-for-suggestions-about-square-numbers (I should add this to my post).

    that diagram is my version of it....

     – PacharapanK Oct 28 '19 at 18:50
  • I think I have to use another thread to post the rest T_T only few photos left. – PacharapanK Oct 28 '19 at 18:51
  • I am also not quite understanding what your final question is, but it appears at a glance that you are discovering some of the recursive nature of sequences of $n$'th powers and differences of consecutive $n$'th powers. Much of this is already well understood and has been for millenia. It can be proved formally with algebra and/or induction. For instance, $(n+1)^2 = n^2+2n+1 = n^2 + n + (n+1)$. This can of course continue, $(n+1)^2 = n^2+n+(n+1) = (n-1)^2 + (n-1)+n+n+(n+1)$ and so forth ad nauseum. Similar identities exist for higher powers. – JMoravitz Oct 28 '19 at 18:56
  • Dear JMoravitz, that's what I want to know... unfortunately, I don't think I understand the terms T_T. I have some more photos to post, do you think I can start a new threat to post the rest of them? – PacharapanK Oct 28 '19 at 19:00
  • You can try using imgur album – helpme Oct 28 '19 at 19:00
  • Thank you I'll try that! – PacharapanK Oct 28 '19 at 19:04
  • It doesn't work... It limits numbers of link I can add, including links pasted from my own upload... – PacharapanK Oct 28 '19 at 19:11
  • Put your physical pieces of paper side by side and take a picture of multiple at once – helpme Oct 28 '19 at 19:15
  • Dear helpme, I just tried that with picture collage, but I have 8 more picture TwT If I snap them together to put every of them in 8 photos, it will be too small to read. I think I'll just use another thread T_T please excuse my inappropriate manner of using webboard. – PacharapanK Oct 28 '19 at 19:20

1 Answers1

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For any sequence $u_n$, define the forward difference $\Delta u_n:=u_{n+1}-u_n$, in analogy with the derivative $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ of a continuous function $f$. Your first finding is $\Delta n^2=2n+1$. We can iterate $\Delta$, e.g. $\Delta^2u_n:=\Delta(\Delta u_n)$ so $\Delta^2n^2=2$. The pattern you've found is $\Delta^kn^k=k!$ for any integer $k\ge0$ (in analogy with $\frac{d^k}{dx^k}x^k=k!$). To prove this, it suffices to note from the binomial theorem that $\Delta n^k$ is a degree-$(k-1)$ polynomial in $n$ with leading coefficient $k$.

J.G.
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  • Thank you J.G.... I'm so sorry I don't understand the signs (T_T; do you think I can have more simple version of them to read, please...? – PacharapanK Oct 28 '19 at 19:02
  • @PacharapanK My first equation tells you what $\Delta$ means in this context. Each of your "layers" goes from a sequence, say $u_n$, to the corresponding $\Delta u_n$. – J.G. Oct 28 '19 at 19:08
  • Does the Δ have a name...? What do we call it? – PacharapanK Oct 28 '19 at 19:10
  • @PacharapanK A forward difference, as I said. Well, it's a first forward difference; $\Delta^ku_n$ is a $k$th forward difference. – J.G. Oct 28 '19 at 19:12
  • Thank you. And what is the called, please? – PacharapanK Oct 28 '19 at 19:13
  • @PacharapanK Oh, $u_n$ is just a common notation for the $n$th term in whatever sequence we're interested in. When I discussed your first example, I took $u_n$ to be $n^2$. – J.G. Oct 28 '19 at 19:14
  • I've posted the rest,and if it's ok with you, could you please take a look and convert it to some specific terms for me again, so that I can search for them. Thank you very much! https://math.stackexchange.com/questions/3412849/continue-about-square-numbers – PacharapanK Oct 29 '19 at 04:42