creigh at o2online.de creigh at o2online.de
Thu Sep 9 03:12:06 CEST 2004

```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)
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

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