# name of mathematical progression
**URL:** https://boards.straightdope.com/t/name-of-mathematical-progression/696518
**Category:** Factual Questions
**Created:** [August 25, 2014, 11:58pm UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518 "2014-08-25T23:58:30Z")
**Posts on this page:** 14
**Page:** 1
### Author:  [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)
#### Post date: [August 25, 2014, 11:58pm UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/1 "2014-08-25T23:58:30Z")
messing around on my calculator today, I discovered a fascinating sequence:
1^3=1^2
1^3+2^3=3^2
1^3+2^3+3^3=6^2
1^3+2^3+3^3+4^3=10^2
1^3+2^3+3^3+4^3+5^3=15^2
etc
I’m sure somebody noticed this before, and had this named after him.
what’s it called?
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### Author:  [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)
#### Post date: [August 26, 2014, 12:53am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/2 "2014-08-26T00:53:10Z")
> [@Enola\_Straight](#):
>
> messing around on my calculator today, I discovered a fascinating sequence:
>
> 1^3=1^2
> 1^3+2^3=3^2
> 1^3+2^3+3^3=6^2
> 1^3+2^3+3^3+4^3=10^2
> 1^3+2^3+3^3+4^3+5^3=15^2
> etc
>
> I’m sure somebody noticed this before, and had this named after him.
> what’s it called?
I’m not sure they have a name (or need one).
The formula for the nth triangular number is T[sub]n[/sub] = ½n(n+1), from this it is easy to show that T[sub]n[/sub][sup]2[/sup] - T[sub]n-1[/sub][sup]2[/sup] = n[sup]3[/sup]
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### Author:  [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)
#### Post date: [August 26, 2014, 12:59am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/3 "2014-08-26T00:59:16Z")
Probably not what you were hoping for, but [A000537](https://oeis.org/A000537).
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### Author:  [@septimus](https://boards.straightdope.com/u/septimus)
#### Post date: [August 26, 2014, 1:12am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/4 "2014-08-26T01:12:37Z")
[Nicomachus’s theorem](https://en.wikipedia.org/wiki/Nicomachus%27s_theorem). It is also called Aryabhata’s Identity, although Aryabhata lived four centuries after Nicomachus.
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### Author:  [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)
#### Post date: [August 26, 2014, 1:13am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/5 "2014-08-26T01:13:15Z")
So, adding up triangular cubes equals triangular squares HAS NO special name?
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### Author:  [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)
#### Post date: [August 26, 2014, 2:41am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/6 "2014-08-26T02:41:00Z")
huh…ninja’ed:(
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### Author:  [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)
#### Post date: [August 26, 2014, 4:57pm UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/7 "2014-08-26T16:57:16Z")
Thanks for the OP. I had not noticed that. Working out why it holds gave me a chance to exercise my long unused proof by induction skills.
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### Author:  [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)
#### Post date: [August 28, 2014, 1:17am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/8 "2014-08-28T01:17:05Z")
Even better, notice that the squares on the right side are the sum of the first n numbers. E.g.
1^3 + 2^3 + 3^3 + 4^3 + 5^3 = (1 + 2 + 3 +4 +5)^2
But I never knew any name for this.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [August 28, 2014, 1:29am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/9 "2014-08-28T01:29:58Z")
That’s what “triangular numbers” means.
.
…
…
…
…
etc.
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### Author:  [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)
#### Post date: [August 28, 2014, 3:09am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/10 "2014-08-28T03:09:55Z")
Here’s the simplest proof I know, though others will surely feel differently:
Consider 1[sup]p[/sup] + … + n[sup]p[/sup] for integer n. Note that this is always given by a (p+1)-th degree polynomial in n with leading coefficient 1/(p + 1), sending 0 to 0 and 1 to 1; furthermore, this polynomial will be symmetric around -0.5 for odd p and antisymmetric around -0.5 for nonzero even p.
Thus, T[sub]3/sub and (T[sub]2/sub)[sup]2[/sup] are both given by 4th degree polynomials with leading coefficient 1/4, sending 0 to 0 and 1 to 1, symmetric around -0.5. This leaves no degrees of freedom for them to differ, and thus they are the same.
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### Author:  [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)
#### Post date: [August 28, 2014, 3:21am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/11 "2014-08-28T03:21:14Z")
Whoops, I made a mistake in editing: I meant to say “Let T[sub]p/sub be 1[sup]p[/sup] + … + n[sup]p[/sup] for integer n”, so I could refer to that name again later.
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### Author:  [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)
#### Post date: [August 28, 2014, 3:38am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/12 "2014-08-28T03:38:11Z")
Also, I think you meant:
Thus, T[sub]3/sub and (T[sub]1/sub)[sup]2[/sup] are both given by 4th degree polynomials …
?
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### Author:  [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)
#### Post date: [August 28, 2014, 3:39am UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/13 "2014-08-28T03:39:01Z")
Yes, good catch!
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### Author:  [@AdamAndEvil](https://boards.straightdope.com/u/AdamAndEvil)
#### Post date: [August 28, 2014, 9:32pm UTC](https://boards.straightdope.com/t/name-of-mathematical-progression/696518/14 "2014-08-28T21:32:40Z")
Yes, but does [.999 = 1?](http://boards.straightdope.com/sdmb/showthread.php?t=32760)
Sorry, couldn’t resist.