[seqfan] Re: How to calculate oblong root?

[Frank Adams-Watters]

Note that the ceiling of this function (sqrt(n+1/4)-1/2) is A000194. The floor is not in the database.

Franklin T. Adams-Watters

-----Original Message-----
From: David Applegate <david at research.att.com>
To: seqfan <seqfan at list.seqfan.eu>
Sent: Sun, Jun 21, 2015 10:57 pm
Subject: [seqfan] Re: How to calculate oblong root?


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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