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

Gerald McGarvey Gerald.McGarvey at comcast.net
Sat Apr 23 21:11:49 CEST 2005

```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

At 11:23 AM 4/22/2005, Creighton Dement wrote:
>Gerald McGarvey wrote:
>
>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
>
>***********
>
>Thank you for that information. Would you care to send it in as a
>comment?
>
>[ Correction to my first message: The "Smith College Diploma Sequence"
>is
>http://www.research.att.com/projects/OEIS?Anum=A057681 ]
>
>I wrote:
>"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)?"
>
>Last year, I sent the Seqfan message, enclosed in "****", at the bottom
>of the page. The sequence property described is:
>
>for m > n:  if  s | a(n) and  s | a(m)  then   s | a(2m - n)
>
>It may be that I am overlooking something obvious as I have only done a
>bit of empircal checking, but I guess that A105660 has this property (if
>Gerald's recurrence relation for A057083 is true, then it is clear that
>that sequence also has the property after checking the first few terms).
>
>Just to point out... my sequence began:
>3,10,27,49,0,-485,-2643,-9602,-26163,-47525,0,470449,
>2563707,9313930,25378083,46099201,0,-456335045,-2486793147,
>-9034502498,-24616714347,-44716177445
>
>a(3) = 49 = 7^2. The next term divisible by 7 is a(11) = 470449 =
>(7^2)(6901). It follows that if the sequence is to have the property,
>a(2*11-3) = a(19) = 9034502498 = 2*(7^2)*(92188801) should be divisible
>by 7, etc.
>
>IMHO, I propose that the above simple property is given a name (which I
>would be more than happy to let Gerald choose...) if it does not already
>have one.
>
>By the way, I just submitted an unrelated sequence, below, which also
>appears to have the property. [ Sorry, I forgot to factor the
>denominator
>  (-x^12-1+2*x^6) = (x-1)^2*(x+1)^2*(x^2+x+1)^2*(x^2-x+1)^2 ]
>
>-1 1 2 2 9 9 4 -4 5 5 18 18 7 -7 8 8 27 27 10 -10 11 11 36 36 13 -13 14
>14 45 45 16 -16 17 17 54 54 19 -19 20 20 63 63 22 -22 23 23 72 72 25 -25
>26 26 81 81 28 -28 29 29 90 90 31 -31 32 32 99 99 34 -34 35 35 108 108
>37 -37 38 38 117 117 40 -40 41 41 126 126 43 -43 44 44 135 135 46 -46 47
>47 144 144 49 -49 50 50 153 153 52 -52 53 53 162 162 55 -55 56 56 171
>171
>%N A000001 Expansion of
>(1-x-2*x^2-2*x^3-9*x^4-9*x^5-6*x^6+6*x^7-x^8-x^9-2*x^13+2*x^12)/(-x^12-1+2*x^6)
>%H A000001 C. Dement, <a
>href="http://www.crowdog.de/13829/home.html">The
>Floretions</a>.
>%H A000001 C. Dement, <a
>href="http://www.crowdog.de/SeqContext/Plush.html">Sequences in
>Context</a>.
>%o A000001 Floretion Algebra Multiplication Program, FAMP Code:
>4kbasesigcycrokseq[+ .25'j - .25'k + .25j' - .25k' + .5'ii' + .25'ij' +
>.25'ik' + .25'ji' + .25'ki' + .5e]. See "Sequences in Context" for
>details on the "roktype" used.
>
>Sincerely,
>Creighton
>
>*****************
>[snip]
>a(1) = 2
>a(2) = 7
>a(3) = 11
>a(4) = 25 = 5*5
>a(5) = 47
>a(6) = 97
>a(7) = 191
>a(8) = 385 = 5*7*11
>a(9) = 767 = 13*59
>a(10) = 1537 = 29*53
>a(11) = 3071 = 37*83
>a(12) = 6145 = 5*1229
>a(13) = 12287 = 11*1117
>a(14) = 24577 = 7*3511
>a(15) = 49151 = 23*2137
>a(16) = 98305 = 5*19661
>a(17) = 196607 = 421*467
>a(18) = 393217 = 11*35747
>a(19) = 786431
>a(20) = 1572865 = 5*7*44939
>a(21) = 3145727 = 13*241979
>a(22) = 6291457 = 347*18131
>
>One could assume (law of small numbers) that the following holds: for m
> > n
>( s | a(n) ) and ( s | a(m) ) -> ( s | a(2m - n) )
>
>For ex. ( 7 divides a(2) = 7 ) and ( 7 divides a(8) = 385 = 5*7*11 )
>  and ( 7 divides a(2*8 - 2) = a(14) = 24577 = 7*3511 )
>and ( 7 divides a(2*14 - 8) = a(20) = 1572865 = 5*7*44939 )
>
>( If this is true, then 13 should divide a(21*2 - 9) =  a(33) )
>
>Do these types of sequences have a name (if so, what would be the best
>way to prove it...)?
>In any case, the above sequence (unlisted in OEIS) is connected with the
>
>Jacobsthal-Lucas numbers
>http://www.research.att.com/projects/OEIS?Anum=A014551 via
>  c(n) +  b(n) + A014551(n+1) = 4*a(n)
>
>(b(n)) =  (5, 19, 29, 67, 125, 259, 509, 1027, 2045, 4099, 8189, ) ;
>unlisted
>(c(n)) = (2, 4, 8, 16, 32, 64, )
>
>Apparently, the above "law of small numbers" holds for A014551 as well,
>ex.  A014551(5) = 31 and A014551(15) = 7*31*151 and
>A014551(25) = 31*601*1801
>It probably also holds for (b(n)) [and trivially for (c(n)) ]
>
>Now, for (Fib(n)), one has ( s | Fib(n) ) -> ( s | Fib(kn) ) which is
>readily seen to be a stronger condition than the above statement.
>
>Proof:
>( s | Fib(n) ) and ( s | Fib(m) ) ->   ( s | Fib(kn) ) and ( s |
>Fib(k'm) )
>
>A simple calculation shows:
>
>F(2m - n) = | 2F(2m - n + 2) - F(2m - n + 3) |
>                 = | -3F(2m - n + 3) + 2F(2m - n + 4) |
>                 = | 5F(2m - n + 4) - 3F(2m - n + 5) |
>                 = | F(n) F(2m-1) - F(n-1)F(2m) |
>
>Thus, setting k' = 2 and k = 1, it follows that s |  F(2m - n),   q.e.d.
>
>
>*********************

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