[seqfan] Re: sequence challenge

Ed Jeffery ed.jeffery at yahoo.com
Thu Nov 24 07:22:41 CET 2011


Alois,

I don't feel right about submitting more sequences until I can make a donation to OEIS as everyone should. I feel bad that NJAS has to ask for donations, and it is up to us users to make sure he doesn't have to do that. Unfortunately, mine will have to be made after the holidays.


Now, about these sequences: if your measure need not be integral, then we might suppose that the matrix need not be integral, since that wasn't made clear previously. If that is the case, then I have an even smaller measure (<9), with the matrix admitting a sequence that is also not in OEIS:

Consider the matrix M=[(1,Sqrt(2),0);(Sqrt(2),0,1),(0,1,0)]. The entries of M total 3+2*Sqrt(2)=5.828... < 9, with the sequence being
{[M^n]_(0,0)}={1,1,3,5,13,25,59,121,273,577,1275,2733,...}, n=0,1,2,...,

with generating function F(x)=(1-x^2)/(1-x-3*x^2+x^3).


I have a barrel full of sequences based on non-integral, symmetric tridiagonal matrices similar to the one above. For example, I have been toying with a sequence of N X N matrices in which the k-th matrix  [A_N]_k, k=0,1,2,..., has entry at 0,0 set to 1, with Sqrt(k) on all super- and sub-diagonal entries and with zeros everywhere else. The tables of these sequences look to be interesting. For N=5, and defining the sequences S_k={[([A_N]_k)^n]_(0,0)}, n=0,1,2,..., such a table begins as
S_0={ 1, 1, 1,  1,  1,   1, ...}
S_1={ 1, 1, 2,  3,  6,  10, ...}
S_2={ 1, 1, 3,  5, 15,  29, ...}
S_3={ 1, 1, 4,  7, 28,  58, ...}
S_4={ 1, 1, 5,  9, 45,  97, ...}
S_5={ 1, 1, 6, 11, 66, 146, ...}
etc., 

but the six 5X5 matrices for the above example are too awkward to write out so I won't give them here. The generating function for the k-th sequence S_k appears to beof the form

F_k(x)=(1-3*k*x+k^2*x^2)/(1-x-4*k*x+3*k*x^2+3*k^2*x^3-k^2*x^4).

The idea, for me, was to take a closer look at the limit sequences as N -> Infinity. Evidently the second and third sequences, S_1 and S_2, converge to known sequences in the limit, but I don't know how to prove this assertion. In particular, it looks like S_2 converges to A126087 (https://oeis.org/A126087) since more and more later terms appear corrected as N increases, but this seems to be the case for all the rows. There are probably similar arguments valid for the resulting column sequences as well, although I haven't checked any of them against the database. 


It seems to me that a table of these limit sequences might be worth looking into, since the result might contain many known sequences but which were previously not known to be related.


Regards,

Ed Jeffery





>Ed, your sequence is a new candidate.

>Please send it to the OEIS.  The measure (sum of the absolute
>values of matrix elements) gives 9.  So it has the same size as
>the one I mentioned.  Your matrix elements are all positive, which
>is a new quality.

>Alois


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