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

Rainer Rosenthal r.rosenthal at web.de
Sat Apr 24 00:01:47 CEST 2010


David Newman wrote:
> Here's the sequence  which led me to A007898 when I searched for
> 1,2,3,3,4,4,7,7,6,6,...
> 
> How many sequences are there of the following sort.
> 
> 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.
> 
> For example:
> 
> There is one sequence of length 1:  +1
> I'll abbreviate +1 to + and -1 to - from here on.
> 
> There are two of length 2:  + + and + -
> There are three of length 3:  + + -,  + - +, + - -
> There are three of length 4:  + + - +, + - + +, + - - +

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.

Cheers,
Rainer
 




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