[seqfan] Re: A139261

[Richard Guy]

It's a bit sweeping to condemn all
``sequences such as k*A######''

Let me give a few hundred frinstances
that ought to be in OEIS but aren't at
present.  See the end of this message
for how you can help.  Note that the
sequences are  k = (q+1-t) times
normalized sequences, which start 0, 1, ...
and if you give only one of the pair

     0, 1, ...   and  0, q+1-t, ...

then you'll miss many real-life
occurrences of the other.

The numbers of points on an elliptic
curve over the finite field  F(q^n)
for  n = 0,1,2,... form a fourth order
divisibility sequence of which the
first few terms are

a(0) = 0, a(1) = q + 1 - t,

a(2) = q^2 + 1 - (t^2 - 2q) = a(1)(q + 1 + t),

a(3) = q^3 + 1 - (t^3 - 3qt) =
      = a(1)(q^2-q+1+(q+1)t+t^2),

a(4) = q^4 + 1 - (t^4 - 4qt^2 + 2q^2) =
      = a(2)((q-1)^2 + t^2)

[[more terms on request, but keen types can
generate them from the recurrence, which is

a(n) = (q+1+t)a(n-1) - (qt+t+2q)a(n-2)+
           q(q+1+t)a(n-3) - q^2a(n-4)  ]]

They can be written as

        q^n + 1^n + alf^n + bet^n

where  q, 1, {alf, bet} = (t+/-sqrt(t^2-4q))/2

are the roots of the characteristic polynomial.

[and also can be written as
         q^n + 1 - an interesting
sequence of polynomials, which will be
recognizable to some]

For the inquisitive who don't already know,
q  is any prime, and  t  is the trace of
Frobenius (Hecke eigenvalue) for the curve
at the prime  q.  Hasse's theorem tells us
that  |t| leq 2 * sqrt(q)  [note that you
can have equality here, since more generally
q  can be a prime power, perhaps an even one]

But keen OEISers will forget that and notice
that values of  t  outside that range still
give divisibility sequences which are not
connected to elliptic curvss, but may well
have manifestations in other walks of life.

So get to work with  q = 2, 3, 5, 7, 11,
13, 17, ... (at least the first dozen primes?)
and  -c leq t leq c  for  c  at least 10.

You'll find a few surprises.  Meticulous
contributors will quote the first few curves
from Cremona's tables that possess those
numbers of points.  Happy generating!  R.

PS. These sequences are only one, but an
important one, of the examples that Hugh
Williams & I give in a hopefully forthcoming
paper on fourth order divisibility sequences.

On Fri, 7 May 2010, Robert G. Wilson v wrote:

> Et al,
>
>    I agree that sequences like this contribute nothing to the knowledge
> base. I also do not like sequences such as k*A######.
>
> Bob.
>
> --------------------------------------------------
> From: "Michael Porter" <ic_designer at verizon.net>
> Sent: Friday, May 07, 2010 1:01 PM
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan]  A139261
>
>> Given that we have A001620 (Decimal expansion of Euler's constant),
>> do we really need A139261 (Triangle read by rows: row n lists the
>> first n digits of the decimal expansion of Euler's constant)?  It seems to
>> be repeating the same data in an unnatural format.
>>
>> - Michael