[seqfan] Re: plastic number base

Graeme McRae graememcrae at gmail.com
Wed Jul 10 03:42:58 CEST 2013


If p is the plastic number (1.324718ish) then 2=10.00001(base p), i.e.
2=p^2+p^-5

Just as the base phi numbers are related to the fibonacci sequence, base
"plastic" numbers are related to the Padovan sequence (A931). The "standard
form" of a number, base phi, has no pair of consecutive 1's.  I would
conjecture that the "standard form" of a number, base "plastic" would have
no 1's separated by fewer than four zeros, because
A931(n-1)+A931(n-6)+A931(n-11)+...=A931(n) or =A931(n-1) for all n.

--Graeme McRae
Palmdale, CA


On Tue, Jul 9, 2013 at 3:39 PM, Alonso Del Arte <alonso.delarte at gmail.com>wrote:

> I'm guessing 1 would just be 1. How would 2 be represented? Would the
> length of 2's representation be typical for small integers up to about
> 10^4?
>
> Al
>
> On Tue, Jul 9, 2013 at 4:31 AM, Dale Gerdemann <dale.gerdemann at gmail.com
> >wrote:
>
> > Hello Seqfans,
> >
> >
> > Suppose we represent numbers in "plastic number base" (like golden ratio
> > base  but using the plastic number instead of the golden ratio). There
> are
> > some numbers that are expressed using a surprisingly small number of
> terms.
> > For example, 10717 requires the following powers of the plastic number:
> > [33, -18, -42, -47, -53, -67]. Here's  a  small list of examplesL:
> >
> >
> > 10717  [33, -18, -42, -47, -53, -67]
> > 12214  [33, 26, -11, -19, -29, -42, -67]
> > 14197  [34, -19, -24, -33, -42, -47, -54, -67]
> > 24914  [36, -16, -24, -33, -39, -47, -65, -72]
> > 57918  [39, -19, -26, -31, -37, -49, -54, -60, -68, -74, -80]
> > 101639 [41, -19, -28, -34, -51, -64, -72, -80]
> > 115836 [41, 34, -16, -31, -41, -54, -63, -70, -76, -85]
> > 192561 [43, 34, -20, -27, -42, -48, -58, -67, -75, -85]
> > 236282 [44, -20, -33, -42, -51, -57, -62, -80, -85]
> > 370925 [45, 39, -22, -32, -42, -50, -59, -73, -79, -87, -93]
> > 414646 [46, -23, -30, -42, -57, -74, -87, -93]
> >
> > Is there any signicance to these numbers?
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
>
> --
> 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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>



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