# [seqfan] Re: [math-fun] You wouldn't believe (fwd)

Richard Guy rkg at cpsc.ucalgary.ca
Sat Dec 20 23:31:50 CET 2008

```Sorry about this, which turns out to be an
unintentional hoax.  Fred Helenius observes
that I managed accidentally to insert the
19-digit prime into the term a(103), which
is only

9311823293177912868609791130160292081658870701

Note that the offset in A005178 is ``wrong''.
This is a ``divisibility sequence'', i.e.,
with the ``right'' offset  a(m)  divides  a(n)
just if  m  divides  n, as in the Fibonacci
sequence.  The way to put it ``right'' is to
define the  n-th member to be the number of
domino tilings of a  4 by n-1 rectangle [# of
matchings of the graph  P4 x P(n-1)] giving
a(2)=1, a(1)=1 (the empty matching of the
empty graph), a(3)=5:
---
| |   ---   ---   | |   ---
| |   ---   | |   ---   ---      and
| |   ---   ---   ---

a(n) = a(n-1) + 5a(n-2) + a(n-3) - a(n-4)

with  a(0) = 0  and  a(-n) = a(n).  Hugh
Williams & I may have a paper within a finite
in which some primes have two ranks of apparition.

R.

On Sat, 20 Dec 2008, Richard Guy wrote:

> Funsters & fansters might be amused by this
> somewhat unlikely coincidence.  I had
> calculated a couple of hundred terms of a
> fourth order recurring sequence [it's
> A005178 in OEIS, if you want details]
> and was looking for the ranks of apparition
> of various primes.  A 19-digit prime factor
> of the 53rd term turned up as a substring
> of the 103rd term!
>
> Have a prime time in  7^2 x 41.    R.
>
> ---------- Forwarded message ----------
> Date: Wed, 17 Dec 2008 09:24:06 -0700 (MST)
> From: Richard Guy <rkg at cpsc.ucalgary.ca>
> To: Hugh Cowie Williams <williams at math.ucalgary.ca>
> Subject: You wouldn't believe
>
> Hugh,
>      before sending the final (??) version
> of the quadric file, I decided I'd do a search
> for larger primes just to see if there were
> any double ranks.  While searching with
>
>               3140540902719737029
>
> which is a factor of a(53), I discovered that it's
> a substring of a(103):
>
> 93118232931779128686097911301602920314054090271973702981658870701
>                                    ^^^^^^^^^^^^^^^^^^^
> R.

```