AW: Re: Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2

[Creighton Dement]

Gerald McGarvey wrote:
> I sent comments regarding the recurrences (I don't have proofs).
> 
> I don't know if the property has a name; if it doesn't, I wouldn't
> know what to call it.
> 
> For the new sequence -1 1 2 2 9 9 4 -4 5 5 18 18 7 -7 ...
> it looks like (assuming the sequence starts with index 0):
> the initial terms are  -1 1 2 2 9 9 4 -4 then
> if n == 2 (mod 6) then a(n) = a(n-6) + 3
> if n == 3 (mod 6) then a(n) = a(n-6) + 3
> if n == 4 (mod 6) then a(n) = a(n-6) + 9
> if n == 5 (mod 6) then a(n) = a(n-6) + 9
> if n == 1 (mod 6) then a(n) = a(n-6) + 3
> if n == 0 (mod 6) then a(n) = a(n-6) - 3
> 
> Sincerely,
> Gerald

Thanks again, Gerald, for your help. 

Concerning the property 
for m > n:  if  s | a(n) and  s | a(m)  then   s | a(2m - n)

Here are the factorizations of the Pell numbers:
[0, 1, (2), (5), (2)^2*(3), (29), (2)*(5)*(7), (13)^2, (2)^3*(3)*(17),
(5)*(197), (2)*(29)*(41), (5741), (2)^2*(3)^2*(5)*(7)*(11), (33461),
(2)*(13)^2*(239), (5)^2*(29)*(269), (2)^4*(3)*(17)*(577), (137)*(8297),
(2)*(5)*(7)*(197)*(199), (37)*(179057), (2)^2*(3)*(19)*(29)*(41)*(59),
(5)*(13)^2*(45697), (2)*(23)*(353)*(5741), (229)*(982789),
(2)^3*(3)^2*(5)*(7)*(11)*(17)*(1153), (29)*(1549)*(29201),
(2)*(79)*(599)*(33461), (5)*(53)*(197)*(146449),
(2)^2*(3)*(13)^2*(113)*(239)*(337), (44560482149)]

Apparently, it appears that the Pell numbers also have the property
(along with numerators of continued fraction convergents to sqrt(2)) .
Moreover, it appears that num/den. of continued fraction convergents to
sqrt(5) have the property: 
[0, 1, (2)^2, (17), (2)^3*(3)^2, (5)*(61), (2)^2*(17)*(19), (13)*(421),
(2)^4*(3)^2*(7)*(23), (17)*(53)*(109), (2)^2*(5)*(11)*(31)*(61),
(89)*(19801), (2)^3*(3)^3*(17)*(19)*(107), (233)*(135721),
(2)^2*(13)*(29)*(211)*(421)]

Hhmm, perhaps someone would like to prove the above... 

Note: in my mind, I liken for m > n:  if  s | a(n) and  s | a(m)  then  
s | a(2m - n) somewhat to the "parallelogram equation" making a normed
space for which it is valid share properties with the plane R^2.  Since
the property holds for the sequence of natural numbers, I naively assume
it possible to picture sequences with the property as behaving "more
like the sequence of naturals" than those that don't (unfortunately, the
name "Hilbert sequence" is already taken:
http://math.bard.edu/math/research/pdfs/MonomialIdeals.pdf )- of course,
such a statement may also tend to overemphasize the property's
importance.

Sincerely, 
Creighton