[seqfan] Re: How to calculate oblong root?

[M. F. Hasler]

Alonso,
do you have a reference for this precise*  definition of "oblong root" ?
(* it is not that precise, due to the fact that you define it only via its
floor and ceiling. You can add any random function with values between 0
and 1 to its floor and still have the same property. In particular you can
take the integer valued function you will find as inverse of the triangular
numbers (argument x 2) and add some small constant, e.g. 0.5)

It would seem more natural to me to define it as the inverse function of x
-> x (x+1) on the half line on which it is increasing.

I think in sequences related to triangular numbers and indexing functions
to triangular tables (the one that gives the row of the n-th element of a
triangle) you should find what you are looking for (up to the factor 2).

Maximilian
Le 21 juin 2015 21:48, "Alonso Del Arte" <alonso.delarte at gmail.com> a
écrit :

> 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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