[seqfan] A method to generate sequences

[Dmitry Kamenetsky]

Hello all,

I have been looking at cellular automata recently. In particular, I found a
method (perhaps new?) that generates some interesting sequences. For example
you begin with 3 cells (x,y,z). At each iteration every cell transforms to
itself plus all its neighbours. So (x,y,z) becomes (x+y, x+y+z, y+z). For
example if we start with (1,0,0) we obtain the following:

(1,1,0)
(2,2,1)
(4,5,3)
(9,12,8)
(21,29,20)
(50,70,49)
(120,169,119)
(289,408,288)
(697,985,696)

Numbers in the first column seem to form A171842, those in the second form
the Pell numbers (A000129) and the third column seem to form A048739. If you
start with (0,1,0) then the second column forms A001333. You can also use a
different number of starting cells. For example, if you start with (1,0,0,0)
then the columns seem to form A005207, A051450, A094292, A056014
respectively. If you have an infinite number of cells, ie (1,0,...,0) then
the first column generates the Motzkin numbers (A001006). Furthermore, we
can use any other arbitrary transformation rule, eg. each cell transforms to
the maximum of all its neighbours plus one. We can also use arbitrary cell
topologies, such as grids (with 4 or 8 neighbourhood system), triangular
lattices, hexagonal lattices and so on.

I am wondering whether this method is new? If it is new then perhaps it can
be used to generate new interesting sequences, or generate old sequences
more efficiently?


Sincerely,
Dmitry Kamenetsky