# [seqfan] Re: The weight puzzle sequence

Tanya Khovanova mathoflove-seqfan at yahoo.com
Tue Aug 24 23:20:28 CEST 2010

```Zeckendorf representation uses non-consecutive Fibonacci numbers. There are many ways to represent a number as a sum of Fibonacci numbers:
13 = 13 = 8 + 5 = 8 + 3 + 2.

Tanya

--- On Tue, 8/24/10, Dan Dima <dimad72 at gmail.com> wrote:

> From: Dan Dima <dimad72 at gmail.com>
> Subject: [seqfan] Re: The weight puzzle sequence
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Tuesday, August 24, 2010, 4:39 PM
> Hi,
>
>
> Just note that lower bound for *a(n)* can be slightly
> improved by
> considering instead of different powers of two the
> Fibonacci numbers (any
> integer can be represented in unique way as sum of
> Fibonacci numbers -
> Zeckendorf representation).
> *a(n) ≥ log2n
> one gets
> **a(n) ≥ loggn*, g - golden ratio
>
>
>
> Best Regards,
> Dan
>
>
>
>
> On Tue, Aug 24, 2010 at 7:57 PM, Tanya Khovanova <
> mathoflove-seqfan at yahoo.com>
> wrote:
>
> > Dear SeqFans,
> >
> > I just wrote a piece for my blog with a problem from
> the 1966 Moscow
> > http://blog.tanyakhovanova.com/?p=269
> >
> > There is a sequence there. I didn't compute enough
> terms to check if it is
> > in the database. I would like if someone can compute
> more terms of the
> > sequence.
> >
> > Here is the relevant part from the blog:
> >  From the 1966 Moscow Math Olympiad:
> >
> >    Prove that you can choose six weights
> from a set of weights weighing 1,
> > 2, ..., 26 grams such that any two subsets of the six
> have different total
> > weights. Prove that you can’t choose seven weights
> with this property.
> >
> > Let us define the sequence a(n) to be the largest size
> of a subset of the
> > set of weights weighing 1, 2, ..., n grams such that
> any subset of it is
> > uniquely determined by its total weight. I hope that
> you agree with me that
> > a(1) = 1, a(2) = 2, a(3) = 2, a(4) = 3, and a(5) = 3.
> The next few terms are
> > more difficult to calculate, but if I am not mistaken,
> a(6) = 3 and a(7) =
> > 4.
> >
> > Best, Tanya
> >
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
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>

```