[seqfan] Re: A007898 ('Connected with Fibonacci partitions')

[David Newman]

It's my fault for not getting the definition right.  Yes, all the partial
sums must be less than 2 in absolute value.



On Fri, Apr 23, 2010 at 8:09 PM, Andrew Weimholt
<andrew.weimholt at gmail.com>wrote:

> On 4/23/10, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> > David Newman wrote:
> >  >
> >  > A= a_1,a_2,a_3,...,a_k has k elements each of which is +1 or -1, and
> a1=1.
> >  > For all positive integers x and y (a_x) (a_y)= a_xy and
> >  > The sum of elements a_i , a_2i, a_3i, ...is never greater than 2 in
> absolute
> >  > value, where i is a positive integer.
> >
> > Sorry, but I don't see, why there should be only four such sequences
> >  of length 6. What's wrong with these six:
> >
> >            1.   + + - + + -   (discrepancy 2)
> >            2.   + + - + - -   (discrepancy 2)
> >            3.   + - + + + -   (discrepancy 2)
> >            4.   + - + + - -   (discrepancy 1)
> >            5.   + - - + + +   (discrepancy 2)
> >            6.   + - - + - +   (discrepancy 1)
> >
> >  They seem to be multiplicative:
> >  a_4 = a_2*a_2 = +1 and a_6 = a_2*a_3.
> >
>
> Please excuse my uneducated guess, but perhaps David meant
> that even the partial sums of elements a_i , a_2i, a_3i, ...are never
> greater
> than 2 in absolute value.
>
> Even though the final sum of sequences 1 and 3 do not
> exceed 2 in absolute value, the partial sums do exceed it.
>
> For example, look at the first 5 terms of sequence 1...
> + + - + + sums to 3, so nevermind that the final " - " brings it back down
> to 2.
>
> David can correct me if I am wrong :-)
>
> Andrew
>
>
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