[seqfan] Re: How to calculate oblong root?

[Alonso Del Arte]

> any x that satisfies
floor(oblongroot(n)) < x < ceil(oblongroot(n))
will also satisfy the definition of oblongroot(n).

I would not want that. I would expect oblongroot to behave like sqrt in
some ways, such as having a more or less continuous graph for positive
numbers, and maybe requiring imaginary numbers to deal with negative
numbers.

Al

On Sun, Jun 21, 2015 at 11:57 PM, David Applegate <david at research.att.com>
wrote:

> That definition of oblongroot(n) is ambiguous.  If n is not an oblong
> number, then any x that satisfies
> floor(oblongroot(n)) < x < ceil(oblongroot(n))
> will also satisfy the definition of oblongroot(n).
>
> Why not just define oblongroot(n)=m, where m satisfies m^2 + m = n,
> that is, oblongroot(n) =  sqrt(n+1/4)-1/2 (and the secondary oblong
> root is -sqrt(n+1/4)-1/2)?
>
> David Applegate               AT&T Labs Research
> Tel:    +1 908 901 2004       Email:  david at research.att.com
>                               Recycle yourself -- be an organ donor
>
> > From seqfan-bounces at list.seqfan.eu Sun Jun 21 21:48:44 2015
> > Date: Sun, 21 Jun 2015 21:47:43 -0400
> > From: Alonso Del Arte <alonso.delarte at gmail.com>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] How to calculate oblong root?
>
> > The principal square root function is a function that is useful not only
> in
> > mathematics but in many sciences. The principal oblong root function is
> > significantly less useful, but once in a while it comes in handy, as in
> for
> > example, the oblong number equivalent of A048761, the smallest square
> > greater than or equal to n.
>
> > If n is an integer of the form m * (m + 1) or m^2 + m with m also an
> > integer, then oblongroot(n) = m. But if n is not of that form, then
> > oblongroot(n) returns some number, possibly irrational, such that
> > floor(oblongroot(n)) * ceiling(oblongroot(n)) is the smallest oblong
> number
> > greater than n.
>
> > Maybe the formula for oblongroot(n) is very easy, but at the moment, I
> > don't see what it is.
>
> > Al
>
> > P.S. The secondary oblong root is a negative integer of the form -(m +
> 1),
> > e.g., -5 is the secondary oblong root for 20.
>
> > --
> > Alonso del Arte
> > Author at SmashWords.com
> > <https://www.smashwords.com/profile/view/AlonsoDelarte>
> > Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
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-- 
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>