About sequences

[Roland Bacher]

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)
from which your properties can probably be read of by considering
the associated characteristic polynomials.

Best whishes,  Roland Bacher


On Mon, Sep 06, 2004 at 08:01:38PM +0100, creigh at o2online.de wrote:
> Please excuse all the postings, below is a IX 'th  property 
> (without proof - although, 
> Limit as n-> infinity of A001353(n)/A001353(n-1) = 2 + sqrt(3). 
> -Gregory V. Richardson (omomom(AT)hotmail.com), 
> Oct 06 2002 has been proven- I was informed by Neil that 
> "jes(n)" =   [1, -4, 15, -56, ...] is (-1)^(n+1)*A001353(n+1)).
> I also fixed a mistake at the end. 
> 
> *************************
> 
> Just found an additional property (VIII) which perplexes 
> me. I also fixed a mistake at the end. 
> 
> ********************* 
> 
> Apparently, none of the four sequences 
> 
> jes seq: 1(0), -4(1), 15(2), -56(3), 209(4), -780(5), 2911(6), -10864(7), 
> 40545(8), -151316(9), 564719(10) 
> 
> les seq: -2(0), 1(1), -6(2), 16(3), -62(4), 225(5), -842(6), 3136(7), -11706 
> (8), 43681(9), -163022(10) 
> 
> tes seq: -1(0), 4(1), -13(2), 49(3), -181(4), 676(5), -2521(6), 9409(7), -35113 
> (8), 131044(9), -489061(10) 
> 
> ves seq: -2(0), 1(1), -4(2), 9(3), -34(4), 121(5), -452(6), 1681(7), -6274(8), 
> 23409(9), -87364(10) 
> 
> are listed at OEIS. However, notice the symmetries (for all n): 
> 
> I: jes(n) + les(n) + tes(n) = ves(n) 
> 
> II: les(2n+1); tes(2n+1); ves(2n+1); 
> ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1; 
> 3*les(2n+1) + 1 = 3*jes(n)^2 + 1 (see IV and VII) are perfect squares 
> 
> III: les(2n+1) divides ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) -
>  
> 1 
> 
> IV: (jes(n))^2 = les(2n+1) 
> 
> V: tes(2n) = A001570(n), sqrt( tes(2n+1) ) = A001075(n) [both are "nice" 
> sequences] 
> 
> VI: ves(2n), ves(2n)/2 do not exist in OEIS, however: sqrt( ves(2n+1) ) 
> = A001835(n) [ "nice" ] 
> 
> VII: sqrt( les(2n+1) ) = A001353(n) [ Comments: 3*a(n)^2 + 1 is a perfect square. 
> ] 
> 
> VIII: les(n) + tes(n) = ves(2+n) + jes(n) 
> 
> IX: lim n |jes(n+1)/jes(n)| = 
>      lim n |les(n+1)/les(n)| = 
>      lim n |tes(n+1)/tes(n)| =
>      lim n |ves(n+1)/ves(n)| = 2 + sqrt(3)
> 
> It follows immediately from properties I + II that... 
> 1^2 (les(1) + 2^2 (tes(1)) = 2^2 (ves(1) - jes(1) - 1) + 1
> 4^2 (les(3))+ 7^2 (tes(3)) = 8^2 (ves(3) - jes(3) - 1)  + 1 
> 15^2 (les(5)) + 26^2 (tes(5)) = 30^2 (ves(5) - jes(5) - 1)  + 1 
> 
> Sincerely, 
> Creighton