A family of sequences

[Richard Guy]

Much of this is already in OEIS, but perhaps some hero can clean
things up quite a bit, since the relations are often not pointed
out, the simplest formulas, generating functions, not always given.

I refer to the `symmetric third-order recurring sequences' whose
chacteristic polynomial is  x^3 - k*x^2 - k*x + 1.  Specifically
those with initial values  0, 0, 1,  but we should perhaps
consider others [for example 0, 1, 1 ?]

As the polynomial has roots  -1  and (k+1\pm\sqrt{(k+3)(k-1)})/2
the sequences are variants of second order ones.  It turns out
that the terms are products of consecutive pairs of terms of
simpler sequences, whose terms may be independently generated
by alternately adding the two previous terms or adding  k-1
times the previous term to the one before that.

k=1  rather trivial 0 0 1 1 2 2 3 3 4 4 ... probably in OEIS

k=2     A001654     here the simpler sequence is the Fibs.

k=3     A109437                   A002530

k=4     A099025                   A136211

k=5     A084158                   A041011

 From here on they don't appear to be in

k=6  0 0 1 6 42 287 1968 13488 ...    1 6 7 41 48 281 329 ...

k=7  0 0 1 7 56 440 4165 31059 ...    1 7 8 55 63 493 556 ...

and more ??  I expect that the idea of the sequence of
orthogonal sequences to a sequence of sequences is well known
to those who well know it.  I.e., the sequence of 1st terms,
of 2nd terms, of 3rd terms, ...  Here the orthog seqs to the
first set are:

all zeroes, all zeroes, A000012, A000027, the pronic numbers,

and then the following seem not to be in, though they're
values of easy polynomials (of degrees 3, 4, etc.)

2 15 44 95 174 287 ...   3 40 165 456 1015 1968 ... etc.

The sequences orthogonal to the simpler sequences are:

A000012, A000027, A000027, A028387, A005563, A123972,

and maybe more --- I haven't checked the next one:
21, 56, 115, 204, 329, ... The degrees of these polynomials
are 0, 1, 1, 2, 2, 3, 3 (this last) and this sequence is
k = 1  which is where we came in!  Thanks in anticipation
of your cleaning this up.    R.




rg> From seqfan-owner at ext.jussieu.fr  Sun May 18 19:10:12 2008
rg> Date: Sun, 18 May 2008 11:08:40 -0600 (MDT)
rg> From: Richard Guy <rkg at cpsc.ucalgary.ca>
rg> To: "Sloane's Dream Team" <editors at seqfan.net>,
rg>         "Neil J. A. Sloane" <njas at research.att.com>, seqfans at seqfan.net,
rg>         Sequence Fans <seqfan at ext.jussieu.fr>
rg> Subject: A family of sequences
rg> ..
rg> I refer to the `symmetric third-order recurring sequences' whose
rg> chacteristic polynomial is  x^3 - k*x^2 - k*x + 1.  Specifically
rg> those with initial values  0, 0, 1,  but we should perhaps
rg> consider others [for example 0, 1, 1 ?]
rg> ...

One could try to track/combine these by adding two tables,
one for generating function x/[(1+x)(x^2-kx-x+1)] in the k-th line
of the table


and another for generating function 
x/(x^2-kx-x+1) in the k-th line, (that is the convolution of 
the previous table with [1,1] along each line):


and then write a loooong comment where this all came from, where
the columns/rows are found etc.

Richard Mathar  www.strw.leidenuniv.nl/~mathar




The next question that arises in search of duplicates is:
Are the two sequences
http://research.att.com/~njas/sequences/?q=id:A104722|id:A59348
the same apart from the initial term?

Richard