[seqfan] Re: Symmetric Linear Recorrence Coefficients?
Max Alekseyev
maxale at gmail.com
Thu Apr 21 11:52:12 CEST 2011
Symmetric coefficients imply that the characteristic polynomial f(z)
can be expressed in terms of a polynomial in t=(z+1/z). That may
potentially lead to finding its roots and thus to an explicit formula
for the sequence.
E.g.:
for A189327, we have
f(z) = z^4 - 2*z^2 + 1 = z^2 * ( t^2 - 4 ) = z^2 * (t - 2) * (t + 2)
for A189328 , we have
f(z) = z^14 + 3*z^13 + 5*z^12 + 5*z^11 + 2*z^10 - 3*z^9 - 8*z^8 -
10*z^7 - 8*z^6 - 3*z^5 + 2*z^4 + 5*z^3 + 5*z^2 + 3*z + 1
= z^7 * ( t^7 + 3*t^6 - 2*t^5 - 13*t^4 - 9*t^3 + 4*t^2 + 4*t ) = z^7 *
t * (t-2) * (t + 2) * (t^2 + t - 1) * (t + 1)^2
etc.
Max
On Wed, Apr 20, 2011 at 5:27 PM, Ron Hardin <rhhardin at att.net> wrote:
> Do symmetric linear recurrence coefficients have significance?
>
> http://oeis.org/A189327
> http://oeis.org/A189328
> http://oeis.org/A189329
> http://oeis.org/A189330
>
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
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