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

[Gerald McGarvey]

I get the following:
A105660: For n > 11, a(n) = - 970*a(n-6) - a(n-12)
A057083: For n > 5, a(n) = -27*a(n-6)

Sincerely,
Gerald

At 06:26 AM 4/21/2005, Creighton Dement wrote:
>Dear Seqfans,
>
>I noticed that the two sequences, below, have some rather obvious
>properties in common as described in the Comments entry (also given,
>below. Note that  "strictly increasing" needs to be replaced by
>"increasing", etc.) of A105660. If the link to
>http://www.crowdog.de/SeqContext/FiveSigns.html given for A105660 is
>followed one is perhaps inclined to suspect that this is some general
>property of a certain class of sequences. I found it aesthetic that
>several of the sequences from the link (which are all interlinked
>through various identities) also have similar properties.
>
>Can someone find more in common between these two sequences (or confirm
>that they belong to a more general class of sequences with those
>properties)?
>
>Thanks and sincerely,
>Creighton
>
>http://www.research.att.com/projects/OEIS?Anum=A057083
>Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of
>1/(1-3*x+3*x^2).
>
>1,3,6,9,9,0,-27,-81,-162,-243,-243,0,729,2187,4374,6561,
>6561,0,-19683,-59049,-118098,-177147,-177147,0,531441,
>1594323,3188646,4782969,4782969,0,-14348907,-43046721,
>-86093442,-129140163,-129140163,0
>
>(=  ((-1)^n)*A105660 ; Smith College Diploma Problem)
>
>and
>
>http://www.research.att.com/projects/OEIS?Anum=A105660
>
>3,10,27,49,0,-485,-2643,-9602,-26163,-47525,0,470449,
>2563707,9313930,25378083,46099201,0,-456335045,-2486793147,
>-9034502498,-24616714347,-44716177445,0
>Name:      G.f. (1-x)(x^2-5x+3)/(x^4-6x^3+13x^2-6x+1).
>
>Comments:  One of a set of interlinked sequences which appear to have
>the property that if a(m) = 0 for some m, then a(m+1), a(m+2), a(m+3),
>a(m+4), a(m+5) are strictly increasing or decreasing and a(m+6) = 0.
>Furthermore, for               this particular sequence it would appear
>that a(m+3) is always even with a(m+1), a(m+2), a(m+4), a(m+5) odd.
>(a(n)) sequence is "ves" in  the link to sequences in context. The
>identity ves = jes + les + tes holds.