[seqfan] A003733

[Richard Guy]

A003733 = 5 * [a(n)]^2, where
a(n) = 0, 1, 19, 319, 5301, 88000, 1460701,
24245719, 402446619, 6680076601, 110880352000,
1840465787401, 30549274537419, 507077165538919
8416803858813901, 139707705280792000,
2318961358994380101, 38491662100532675119,
638910193787827477219, 10605056093962494817201,
176029770458537601408000,
2921859140879620328638801,
48498960243504244160458819,
805017980433134457506461519,
13362223552152134930686686501,
221795068678617751944002200000,
3681501981923485805226349963501,
61108017070233359898822099688519,
1014311487156912505215344900691819,
16836216298689637455787203673021801,
279458709524022209298400122403168000,
4638641422949057945221519706079754201,
76995253743735136577129210921872164219,
1278018402054713951550586514688288368119,
21213398963872932558548756605573832423101,
352114097009050465135872561891492484312000, ...

seems not to be in OEIS (but then, as my mother
always told me, I'm not a good looker).  It
is a divisibility sequence with recurrence relation
a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4),
from which the assiduous reader will obtain
the generating function and a formula for a(n).
I must be missing something, 'cos it's given in
section 8.1 of Per Hakan Lundow, Enumeration of
matchings in polygraphs, in connexion with the
graph C_5 x P_2n.

Hugh Williams is writing one or more papers
about fourth-, sixth-, eighth-, ... order
divisibility sequences.  This is a nice
example in which some primes have two ranks
of apparition:

19 has 2 & 9
29 has 3 & 14
41 has 7 & 41 (41 divides the discriminant) 
71 has 7 & 18
79 has 8 & 13
241 has 6 & 15
251 has 9 & 21
281 has 10 & 47
8329 has 7 & 12  !!

I've only looked at the first 50 terms, so
there are no doubt lots of others to be found.
In fact I believe that Hugh can prove that
the condition for such has to do with the
quadratic character modulo the various prime
factors of the discriminant.

Some of the sequences in OEIS are divisibility
sequences having primes with as many as four
ranks of apparition, but I don't think that
this has been revealed.   R.