Talk:Triangular number/Archive 1

Archive 1

666

666 is one of the most famous triangular numbers, wherever its fame comes from, and it's known as the "Number of the Beast" first from the Bible, only secondarily from numerology. Many people would have no idea what its numerological significance is, but would know what it meant in the Bible. — Preceding unsigned comment added by Jacquerie27 (talk • contribs) 16:38, 24 April 2003 (UTC)

Ouzo Cross

Triangular numbers are presented in "The Ouzo Prophecy," and provide the basis for the construction of the Ouzo Cross.

Robert Merlin Evenson/Church of Ouzo

bobevenson@yahoo.com. — Preceding unsigned comment added by 208.30.83.3 (talk) 22:25, 25 July 2005 (UTC)

I have added some links to a video Podcast and they were remove

Hi everyone I have added some links to a video podcast that I own. I think they are a nice addition to wikipedia please look at them and express you oppinion here , judge for yourself if the links are really useful or not to wikipedia.

• [1] Video PodCast by http://www.isallaboutmath.com/

If any of you think they are valuable to wikipedia then feel free to add them back in the external links.

Regards SilentVoice 03:22, 22 January 2007 (UTC)

If your material is sincerely useful, please consider adding to other wikis that are more inclusive, such as wikinfo and wikiknowledge.--69.87.194.231 23:21, 11 February 2007 (UTC)

Right angle triangle

Triangular numbers may also be graphically represented by a right angle triangle. Percussim 09:15, 25 April 2007 (UTC)

Reason for cleanup?

I don't know why this article was tagged for clean up. Having read it a couple of times, the only thing that to me seemed to need cleaning up was the Maple example, which I decided to remove (right decision, wrong decision, I'm not sure). Any reader who has Maple ought to be able to figure out how to use the formulas given here to instruct Maple to calculate these numbers. And if not, he can always look it up in the OEIS. PrimeFan 23:17, 25 February 2007 (UTC)

there are so many theorems and interesting relations between tri#'s and other #'s that it would be difficult to present all of it without the article reading like a long trivia section. i removed that eye-sore table in the introduction.Essap 02:30, 5 May 2007 (UTC)

Floor of square root

the part that said n = the floor of sqrt(2x) is misleading at best. if you put for example x=8, you get n=4, but 8 is not the 4th triangular number. i removed it becuase of that.Essap 02:36, 17 May 2007 (UTC)

Not quite perfect

http://en.wikipedia.org/wiki/Perfect_number should probably be read carefully, 10 isn't a perfect number and yet is a triangular number. — Preceding unsigned comment added by 90.240.83.49 (talk) 22:52, 28 June 2007 (UTC)

Triangle root formula

It is good to see that others have found what I found about 10 years ago: a triangle root formula. Once I discovered it, I saw that I could extract the root of any polygonal number in existence. It's the 3D numbers that now have me stumped. I know how to generate the tetrahedral and octahedral numbers but have yet to find a way to extract their roots. Any help here is welcome. :) FYI: I am the moderator for http://forums.delphiforums.com/figurate/start — Preceding unsigned comment added by R3hall (talk • contribs) 23:28, 30 June 2007 (UTC)

"Whenever a triangular number is divisible by 3, one third of it will be a pentagonal number"

not true, for instance 6, 21 --86.143.232.149 12:26, 10 July 2007 (UTC).

Not quite perfect continued

The claim is made in this article that all even perfect numbers are also triangular, but the second (third) perfect number, 28, isn't triangular, as far as I can figure out. am I missing something?

Samois98 04:15, 10 July 2007 (UTC)

28 is the 6th (or thereabouts... short term memory problems) triangular number.66.216.172.3 16:51, 10 July 2007 (UTC)

"Triangular numbers ... describe numbers of balls that can be arranged in a triangle."

Can someone elaborate on this in the article? It isn't clear to me why I can't fit a non-triangular number of balls into a triangle, or just what sort of balls-into-triangle sense is meant. 66.216.172.3 16:51, 10 July 2007 (UTC)

Yes, this should be made clearer here. Take a look at the graphic at Figurate number to see what kind of triangles is meant. Cheers, Doctormatt 07:52, 14 July 2007 (UTC)

That's what I (eventually) assumed it meant. I'm not sure what the a simple way to describe that would be, other than maybe linking to Figurate number. Does anyone have a good idea? I'll think about it and see if I come up with anything. 207.103.181.5 16:40, 17 July 2007 (UTC)

Games: Here's an image from bowling illustrating $T_{4}$: Ten-pin bowling#Pins. And here's one from pool illustrating $T_{5}$: Eight-ball#Equipment. /84.238.113.244 (talk) 21:58, 16 March 2008 (UTC)

Confusing Wording

"A triangular number is the number of dots in an equilateral triangle evenly filled with dots." What does it mean for the triangle to be evenly filled versus unevenly filled? Why not just "filled"? —Preceding unsigned comment added by 71.193.118.77 (talk) 03:36, 3 March 2009 (UTC)

Formula

I added the simplest formula for finding any given triangle number I could come up with (surprised it wasn't on here) and a little bit of trivia about the formula. I'll be checking back later (this knowledge is the result of a 14 hour plane ride! I want to make sure it's respected) although I may be under a different IP than this one. --24.245.11.103 12:31, 15 May 2007 (UTC)

It was on there. In the second line. —David Eppstein 15:40, 15 May 2007 (UTC)

Oops, I stopped reading that line after the big E looking thing. Sorry, mate! --134.84.5.63 18:43, 15 May 2007 (UTC)

In the initial formula at the top of the page, there seems to be an incorrect deduction, only in the final equation with the reduction of the n squared plus n all over 2, down to (n+1)/2, unless I'm missing a symbol or something (font?). Please check. RS — Preceding unsigned comment added by 151.201.43.241 (talk) 02:44, 2 July 2008 (UTC)

In the mathml it is (n+1) "\choose" 2. I have no clue what \choose is, and am disappointed to have this definition not rendered in the article and also not explained. Is it incorrect or simply impenetrable? 64.127.105.60 (talk) 22:11, 8 June 2009 (UTC)

The symbol \choose is a LaTeX symbol that is used to type e.g. binomial coefficient. There is a link in the following text to the article about binomial coefficient with its meaning and definition explained. It is not explained in this article because it's out of its scope. The formula is correct. With proper image rendering you should see something like this. --Tomaxer (talk) 00:40, 9 June 2009 (UTC)

It wasn't apparent that the surrounding parentheses had any specific meaning, when they are used just adjacently to indicate grouping. Essentially my annoyance is this: most of the articles on mathematical concepts on wikipedia, even ones on familiar and simple topics like triangular numbers, are often expressed in relatively impenetrable math jargon. Wikipedia is an encyclopedia for everyone, not just mathematicians. The articles on various taxonomical groups for example are readable by anyone with a casual acquaintance with biology, or even just animals around them, while the mathematics articles are not. Consider the article on the binomial coefficient which never answers things how it behaves to a person other than a mathemtician. How about a pratical example or a brief discussion in what the machinery encapsulates. In this article, not a single paragraph manages to stay away from spikey jargon. The first two sentences make an attempt and then quickly descend into using terminology of nth, but even this is half-hearted, since triangular numbers are not dots. The only part of this page that is readily accessible is the image, which is a shame. It is a simple topic. JoshuaRodman (talk) 06:04, 10 June 2009 (UTC)

Formula wrong?

Shouldn't:

$T_{n}=1+2+3+\dotsb +(n-1)+n={\frac {n(n+1)}{2}}={\frac {n^{2}+n}{2}}={n+1 \choose 2}$

actually be:

$T_{n}=1+2+3+\dotsb +(n-1)+n={\frac {n(n+1)}{2}}={\frac {n^{2}+n}{2}}=n{n+1 \choose 2}$

At the top of the article? DMcMPO11AAUK/Talk/Contribs 21:32, 13 October 2009 (UTC)

No. See binomial coefficient. The thing at the end with one number over another is not a fraction. As the sentence immediately after the formula in the article explains. —David Eppstein (talk) 21:37, 13 October 2009 (UTC)

Handshake Problem

The 'handshake problem' is not solved by this formula. It would be, if everyone was to shake their own hand. (MKC)

The correct handshake formula is similar see below:

$H_{n}={\frac {n(n-2)}{2}}={\frac {n^{2}-n}{2}}$

— Preceding unsigned comment added by Myesac (talk • contribs) 14:57, 10 January 2008 (UTC)

I have removed the statement --Luca Antonelli (talk) 20:25, 14 February 2008 (UTC)

I shall reintroduce the comment on "handshake problem". It was correct. Above post and formula is wrong. Number of handshakes with $n$ persons is $T_{n-1}$. Number of handshakes with $n+1$ persons is $T_{n}$. /84.238.113.244 (talk) 22:06, 16 March 2008 (UTC)

$H_{n}={\frac {n(n-1)}{2}}={\frac {n^{2}-n}{2}}$ is correct for n people shaking hands. Explanation: each person must shake hands with the other n-1 people, resulting in n(n-1). The factor of two results because of the commutivity of a handshake (e.g. Bob shakes Tom's hand so Tom doesn't have to shake Bob's hand).--Loodog (talk) 02:50, 2 July 2008 (UTC)

The link provided next to the sequence of triangular numbers shows a sequence that starts at 0, whereas the sequence on Wikipedia is shown starting at 1. I'm sure that $T_{1}=1$ but it seems odd that it starts at zero in the link. The handshake problem would start at zero because one person can't shake hands with themselves. I'm probably just confused, man. Measurements (talk) 21:34, 16 March 2011 (UTC)

Triangular Numbers in Nature

In the Janet Left Step Periodic Table (see second illustration at the Wiki page for Alternative Periodic Tables) periods end with the alkaline earths (helium being similarly s2 electronic configuration here). It had already been determined that every other alkaline earth atomic number (4,20,56,120) was identical to every other tetrahedral number from the Pascal Triangle. The triangular numbers also figure in the periodic relation. Counting leftwards, in the Janet table, from the alkaline earths (including 0 for no movement) always lands you on positions within periods where the quantum number ml=0. —Preceding unsigned comment added by 71.127.246.177 (talk) 05:20, 23 March 2011 (UTC)

A Vote for Zero (who is my hero)

Some authors consider 0 to be a triangular number. Specifically, Sloan's On-Line Encyclopedia of Integer Sequences lists 0 as the first triangular number ( http://oeis.org/A000217 ) , and Sloan's is the most common authoritative source for this kind of data. Additionally, the Wikipedia pages for "square triangular number" and "squared triangular number" both indicate that 0 is a triangular number, so there is currently inconsistency in the pages here.

I'm not here to argue for or against the inclusion of 0 (though I admit it seems intuitive to me that it would be as much a triangular number as it is a square number), but if this is not the standard definition we list, it should at least be mentioned that people can and do include 0 in the definition.

There are benefits and detriments to making 0 part of the standard definition. Benefits include consistency with Sloan's resource and definitions, as well as the increased generalizability of triangular numbers to useful situations where 0 is an important element: if 0 is included, then every complete graph (including the "trivial graph" with one vertex and zero edges) contains a number of edges corresponding to the n-th (from zeroth) triangular number. The detriments are mainly the need to reformulate some of the trivia as listed in the "other properties" section: specifically, the "digital root trick", the "ax+b trick", and the "sum of reciprocals" will each need to exclude 0. (I don't think the digital root or ax+b are particularly interesting or mathematically useful, but the reciprocal one is.)

On the other hand, you can extend a few of those results to include 0 now (so every Natural, including 0, is the sum of exactly three triangular numbers, rather than positive integers only are the sum of at most three triangular numbers).

Anyway, the fact that some definitions include 0 should probably be added. (And yes, I am a set theory / CS geek. Very sorry.) --24.26.130.82 (talk) 22:03, 14 June 2011 (UTC)

New proof for sum of two consecutive triangular numbers a perfect square

I have found a new proof for sum of two consecutive triangular numbers resulting into a perfect square : A triangular no.: n(n+1) /2 = n^2+n /2 The next triangular no.: (n+1)^2+n+1 /2 So, sum of two consecutive triangular no.: n^2+n+(n+1)^2+n+1 /2 = n^2+n+n^2+1+n+1/2 = 2(n^2)+2n+2 /2 = n^2+n+1 Since (x+1)(x+1) = (x+1)^2 = x^2+2x+1

Therefore, sum of a triangular no, n, and the next triangular no., results into the square of (n+1). WORTH ADDING, ISN'T IT? 117.226.211.98 (talk) 16:39, 1 January 2013 (UTC)

This isn't really any different from the proof in the article, except you've considered the nth and (n+1)th triangular numbers instead of the (n-1)th and nth. Hut 8.5 16:52, 1 January 2013 (UTC)

Triangle triangle numbers?

How much reasearch has be done around the function n=T(1)+T(2)+T(3)+T(4)+T(5)+T(6)...n, the sum of thefirst n triangler numbers? What is the formula for this, and what is the sum of it's recopicals? Robo37 (talk) 14:25, 10 August 2011 (UTC)

The sums of the triangular numbers are the tetrahedral numbers. Hut 8.5 14:34, 10 August 2011 (UTC)

Oh yeah, how stupid of me. Thanks anyway. Robo37 (talk) 10:40, 11 August 2011 (UTC)

Pascal's triangle can be helpful if you want to find n=T(1)+T(2)... and n=Tetra(1)+Tetra(2).... and so on. 117.226.228.95 (talk) 16:29, 2 January 2013 (UTC)

0 triangular number?

Empty sum says that 0 is a triangular number, well, is it? Moberg (talk) 00:10, 30 November 2011 (UTC)

Yes. Anders Kaseorg (talk) 23:06, 3 August 2012 (UTC)

Is 0 the 0th triangular number or the 1st triangular number? 117.226.228.95 (talk) 16:30, 2 January 2013 (UTC)

It's the 0th. Plug n=0 into the formula given in the article and you get 0. Hut 8.5 19:33, 2 January 2013 (UTC)

An elegant property#2 and #3

If n is a triangular no [I will refer to them as tri no. from now], then 25n + 3 is also a tri no! Lets see how:

25((n^2+n)/2) + 3 = 25n^2 + 25n + 6 /2= (25n^2 + 20n +4) + (5n + 2) /2. Now: (25n^2 + 20n +4) + (5n + 2) /2 or, (25n^2 + 10n + 10n +4) + (5n + 2) /2 or, ((5n(5n + 2) + 2(5n + 2)) + (5n + 2)/2 or, (5n + 2)(5n + 2) + (5n + 2)/2 or, (5n + 2)^2 + (5n + 2)/2 Wow! We found out that every nth tri no. when multiplied by 25 and added 3, it produces the (5n + 1)th tri no! For eg: 1st tri no: 1*25+3 = 28 = 7th tri no!

If n is a triangular no, then 8n + 1 is a square no! Lets see how:

8((n^2+n)/2) + 1 = 8n^2 + 8n + 2 /2= 4n^2 + 4n + 1. Now: 4n^2 + 4n + 1 or, 4n^2 + 2n + 2n + 1 or, 2n(2n+1) + 1(2n+1) or, (2n+1)^2 Wow! We found out that every nth tri no. when multiplied by 8 and added 1, it produces the (2n + 1)th square no! For eg: 1st tri no: 1*8+1 = 9 = 3rd(1*2+1) square no!
Note: Since, (2n+1)^2 = 4(n^2 + n) + 1, the square produced would always a odd, as would be its square root. 117.226.159.194 (talk) 11:59, 15 January 2013 (UTC)