NEW SEQ?

[Paul D. Hanna]

Zak (and Seqfans), 
    Starting with {0,1} as in your example, your table appears to be new
to the OEIS. 
Here are 2 related sequences. 
  
The number of terms in each row appear to be: A052937  
2, 3, 6, 13, 30, 71, 170, 409, 986, 2379, 5742, 13861, 33462, 
G.f.: (2-3*x-x^2)/(1-x)/(1-2*x-x^2) 
   
Row sums appear to be: A024537  
1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 
G.f.: (1-x-x^2)/((1-x)(1-2x-x^2))
  
Note the similarity of the g.f.s. -- see other nice formulas in A052937,
A024537. 
 
Zak, do your calculations verify that these sequences coincide? 
    Paul 
 
On Fri, 17 Mar 2006 20:49:45 -0800 (PST) zak seidov <zakseidov at yahoo.com>
writes:
> Rows of Binary Sequences
> 
> We start with any seed sequence
> (of any length and structure), e.g.:
> {0,1}
> 
> and proceed:
> {0,1,1}
> {0,1,1,1,0,1}
> {0,1,1,1,0,1,1,0,1,1,0,1,1}
> {0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1}
>
{0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,
1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,1}
> ......................................................
> 
> The rule is 
> "Put in  the (binary) sum of neighbors between them!":
> {0,0}=>{0,0,0},{0,1}=>{0,1,0},
> {1,0}=>{0,1,0},{1,1}=>{1,1,0,1}.
> 
> My Q's:
> 1. Is it known?
> 1a. If yes, please give Ref/Cf.
> 1b. If not:
> 2. Is it interesting?
> 2a. If yes, what can be said about patterns, cycles,
> dependence on seeds etc.
> 2b. If not, please ignore it.
> Thanks, Zak