About sequences

[creigh at o2online.de]

Roland Bacher wrote:
******************
Dear Creighton, 

your sequences jes,les,tes and ves seem to be recursively defined by 

jes(n+1)=-4*jes(n)-jes(n-1) 
les(n+1)=les(n-1)+jes(n) 
ves(n+1)=les(n-1)-jes(n-1)+tes(n-1) 
tes(n+1)=les(n-1)+3*jes(n) 

Plus initial conditions for n=0,1. 

They satisfy thus all linear recursion relations (by elimination) 
from which your properties can probably be read of by considering 
the associated characteristic polynomials. 

Best whishes, Roland Bacher 
******************

Thanks again for these formula. My guess is that choosing different initial conditions 
would  lead to an infinite number of "les", "jes", "tes", and "ves" sequences 
with similar perfect square properties. Whether this be the case or not, 
using "FAMP" I appear to be able to generate an infinite number of "les", 
"jes", "tes", and "ves" sequences with similar 
properties. Surprisingly to me, this was not difficult at all.
The formula for the original four sequences was:

x = 'i - 'k + i' - k' - 'jj' -'ij' - 'ji' - 'jk' - 'kj'
y = .5( 'kk' + 'ij' + 'ji' + 1)

"ves(n)" = ves( (xy)^n ), "tes(n)" = tes( (xy)^n ), etc. 

CLAIM: Let m be any natural number. Define 
x(m) = 'i - 'k + i' - k' - m('jj') -'ij' - 'ji' - 'jk' - 'kj'

Then 
"ves_m(n)" = ves( (x(m)y)^n ), "tes_m(n)" = tes( (x(m)y)^n ), etc.
give sequences with similar properties, in particular 
ves_m(2n+1) is a perfect square and m divides ves_m(n)
for all n. 

Ex. m = 11 ->
( ves_11(n) ) = -22, 121, -484, 5929, -287254, 2686321, -14961892, 7623121
( tes_11(n)  ) = -1, 4, 347, 12769, -63781, 481636, 9334079, 118744609
( les_11(n) ) = -12, 81, -1776, 1296, -141672, 893025, -28947252, 66194496
( jes_11(n) ) = -9, 36, 945, -8136, -81801, 1311660, 4651281, -177315984

I note at least 3 things:
1. It is perhaps peculiar that negative signs present in the sequences 
appear to be distributed in a more complicated manner in regards to jes 
and tes, but not for jes or ves for higher m- the signs of ves
are apparently always of the form (-, +, -, +, -, ...) 

2. The possibility of introducing a "freak" turn of events by deliberately choosing 
an m and an odd n such that, for ex., jes_m(n) is a perfect 
square. This happens above at  m = 11 and n = 1. We get the additional 
"bonus" relation 6^2 + 9^2 + 2^2 = 11^2. 

3. (Perhaps this is a groundless statement) In order to produce recurrence sequences 
of the type encountered here, one apparently
needs at least two initial values. Since we are constantly varying only 
one parameter, i.e. m, there could (or must) exist another
"hidden" parameter, similar to m,  in either x or y.   

Sincerely, 
Creighton