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

[Andrew Weimholt]

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