[seqfan] An apology

[Richard Guy]

Dear All,
          Neil will get this thrice, several others
twice.  I have an infinite amount of information
to impart, and I don't have the time & energy to
do it.  More importantly, I don't have the expertise
to put it into the approved form, and minimize the
effort needed on the part of others.

          I'll try to stop this at some reasonable length,
but I fear the worst.  Advice on how to proceed, and who
to do it with, is welcome.

          Much of what I have to say at present concerns
divisibility sequences.  The classical example is the
Fibonacci numbers which exhibit, amongst other things,
the problem of displaying two-way infinite sequences

    ...,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,...

The property of interest here is that  a(m)  divides
a(n)  if  m  divides  n.

          The Fibs satisfy a second order recurrence.
I confine myself for the time being to fourth order
ones.  A fair amount of (even possibly complete)
generality is given by the characteristic polynomial

       x^4 - bx^3 + (1/4)(b^2 - c + 8)x^2 - bx + 1

which factors into

      (x^2  =  rho1 x  +  1)(x^2  =  rho2 x  +  1)

where  rho1, rho2  are  (b +/- sqrt c)/2  and
c == 0 or 1 mod 4  according as  b == 0 or 1 mod 2.

The simplest examples include  b = 2,  c = 12, 16, 20, ...
of which the third has indeed recently been added to
OEIS as A138573 by Tony Noe, possibly as the result
of some previous emission of mine.  Note that a
recurrence relation and formula are given, but much more
could be said.  It raises the possibility of giving
factorizations of those (many) sequences where this
is of interest.  Note that the primes in fourth order
sequences may have two ranks of apparition.  Which
ones will be the subject of a forthcoming paper by my
colleague, Hugh Williams.  E.g., in A138573,  13  has
ranks of apparition  6 & 7,  and  17  has ranks of
apparition  8 & 9.

It's easy to calculate many terms of such sequences
and someone may like to get to work, for example with
b = 2, c = 16, which is (E&OE - all my work is by
hand, and, as Comrie said, forms an uncomfortable
trap for the unwary plagiarist)

...,12,4,2,1,0,1,4,12,31,80,211,552,1444,3782,9901,25920,...

Here is a start on  b = 2, c = 12, with a little more
detail:

  0       0
  1       1
  2       2
  3       3
  4       8 = 2^3
  5      19
  6      42 = 2 * 3 * 7
  7      97
  8     224 = 2^5 * 7
  9     513 = 3^3 * 19
10    1178 = 2 * 19 * 31
11    2707  p?
12    6216 = 2^3 * 3 * 7 * 37
13   14275 = 5^2 * 571
14   32786 = 2* 13^2 * 97
15   75297 = 3 * 19 * 1321
16  172928 = 2^7 * 7 * 193
17  397153  p?
16  912114 = 2 * 3^3 * 7 * 19 * 127
19 2094787  p?
20 4810952 = 2^3 * 19 * 31 * 1021

Here are the ranks of apparition for the first few primes
(the law of repetition is of interest as well as the law
of apparition -- for generalization of the Lucas-Lehmer
theory, see Hugh Williams's book on Edouard Lucas)

prime  2  3  5   7   11  13   17   19 23 29 31  37 ...
r app  2  3 13  6&8  60  14  288  5&9       10  12 ...

[60 obtained by working mod 11;  288 is a wild guess]

There's much, much more, with cubic, sextic, octic, ...
recurrences, and even one of order 32 (A139400 --- tho
it doesn't say so!)  And did you know that

              A003733 = 5 * (A143699)^2
             A003751 = 5^3 * (A004187)^4  ??

so little done, so much to do, ...  I wonder if
anyone's still awake ... ?      R.