# [seqfan] Divisibility sequences

Richard Guy rkg at cpsc.ucalgary.ca
Fri Mar 27 16:42:29 CET 2009

There are infinitely many of these, a few
already being in OEIS, though not always
indicated as such.  I don't have the energy,
endurance or expertise to get any amount of
them into acceptable shape for OEIS, and Neil
is already overwhelmed.  If anyone else is
interested (Tony Noe wd be an excellent
candidate) I could send them several long
TeX files of inchoate, incompetent and
incomplete ramblings and calculations, and
suggestions for further work.

Here's a small sample:

A003733 = 5*(A143699)^2
A003751 = 5^3*(A004187)^4
A003753 = 4*(A001109)*(A001353)^2
A003755 = (A001109)*(A001906)^2

These last two emerged after a complete
analysis of all such fourth order
recurrences by my colleague Hugh Williams.
As they involve spanning trees of graphs,
there are probably combinatorial proofs
as well.

Here's a variant of  A003735  (which is
NOT a divisibility sequence) with the same
recurrence relation, but different initial
conditions, which I believe not to be in
OEIS:

0, 1, 44, 1833, 76208, ...

It is the case  b = 44, c = 1536  in the
following parody of part of Hugh Williams's
theory.

Characteristic polynomial

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

assume that  a_{-n} = a_n  and take  a_0 = 0,
a_1 = 1,
a_2 = b = \alpha_1+\beta_1+\alpha_2+\beta_2,
then

a_3 = \frac{3}{4}b^2+\frac{1}{4}c+2 =
\alpha_1^2+\beta_1^2+\alpha_2^2+\beta_2^2 +
2(\alpha_1+\beta_1+\alpha_2+\beta_2)+1

and I cd clutter you up all the way to a_{10},
but anyone can work them out who wants to.

The case  b = 2, c = 20  has recently been
added to OEIS as A138573 by Tony Noe,
possibly as the result of some previous
emission of mine.

b = 19, c = 205  is  A143699   (cf.  A003729)

sqrt(A003739 / 5) = (A001906)*(b = 15, c = 105)

b = 1, c = 33  is  A003757

...  but I've probably already alarmed the
moderator by now.   Best to all,   R.