[seqfan] Re: How to calculate oblong root?

[David Applegate]

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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