[seqfan] Re: Spanning trees in the cube of a cycle (A005288)

[Neil Sloane]

A friend just sent me a copy of the Kleitman-Golden paper.

It doesn't solve Brendan's problem, but it has several other related
sequences.

One thing that puzzles me about the Baron et al 1985 Fib Quarterly paper
that is the sole reference for A005822
is that although it has many sequences in it, there is no other reference
to this paper in either the EIS nor the OEIS .
And there is no copy of the paper in the binder with the source
material for sequences A5000-A5999.
I admit I am the author-of-record for A005822, but this evidence suggests
that it may have
arrived in my office in a brown envelope with no return address...

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sat, Jan 25, 2020 at 2:24 AM Neil Sloane <njasloane at gmail.com> wrote:

> It is possible that the article
>
> D. J. Kleitman and B. Golden. "Counting Trees in a Certain Class of
> Graphs." Amer. Math. Montlhy (1975), pp. 40-44. Vol 82, Number 1,
>
> may be relevant, but my Rutgers library login has suddenly  stopped
> working and it 2am here .
> Could someone get me a copy?
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Sat, Jan 25, 2020 at 2:03 AM Neil Sloane <njasloane at gmail.com> wrote:
>
>> 1.  I will ask Simon Plouffe if he can remember anything about this
>>
>> 2.  I checked my old files, but did not find anything on an initial search
>>
>> 3.  If all else fails, I suggest we change the definition of A005822 to
>> the generating function
>> (1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1)
>> and then Brendan you could submit your version that matches the words of
>> the present definition as a new sequence.
>>
>> But let's wait to hear from Simon.
>>
>> Simon, Can you throw any light on A005822 (aka M1243)?
>>
>>
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>>
>> On Fri, Jan 24, 2020 at 11:58 PM Brendan McKay <Brendan.McKay at anu.edu.au>
>> wrote:
>>
>>> Yes, sorry, A005822.   Edited below. Brendan.
>>>
>>> On 25/1/20 1:26 pm, Sean A. Irvine wrote:
>>> > You mean A005822 ...
>>> >
>>> > On Sat, 25 Jan 2020 at 14:31, Brendan McKay <Brendan.McKay at anu.edu.au>
>>> > wrote:
>>> >
>>> >> It has been noticed on Mathoverflow that something is amiss with
>>> A005288
>>> >> "Number of spanning trees in third power of cycle".
>>> >>
>>> >> A5822(n) = 1, 1, 2, 4, 11, 16, 49, 72, 214, 319, 947, 1408, 4187,
>>> >>             6223, 18502, 27504
>>> >> offset: 1,3
>>> >>
>>> >> The simplest proof that there's a problem is that there is no graph on
>>> >> 3 vertices with 2 spanning trees.
>>> >>
>>> >> I take "third power of cycle" to mean a graph whose edges join
>>> vertices
>>> >> arranged
>>> >> in a cycle if they are at most 3 apart.  Up to 7 vertices it is a
>>> complete
>>> >> graph,
>>> >> so the number of spanning trees is n^(n-2).  Starting at 7 vertices
>>> it is a
>>> >> regular graph of degree 6.
>>> >>
>>> >> By direct computation using the matrix tree theorem, I get actual
>>> values
>>> >> T(n) = 1, 1, 3, 16, 125, 1296, 16807, 82944, 412164, 2035220, 9864899,
>>> >>       47579136, 227902597, 1084320412, 5134860060, 24207040512,
>>> >>       113664879137, 531895993344
>>> >>
>>> >> I'm sure there is some relationship between A5288(n) and T(n) since
>>> >> for n >= 7 and as far as I computed T(n) is a multiple of A5288(n).
>>> Here
>>> >> is the ratio T(n)/A5822(n) starting at n=7:  1152, 1926, 6380, 10417,
>>> >> 33792,
>>> >> 54431, 174244, 277530, 880128, 1390073, 4375872
>>> >>
>>> >> I conjecture  that T(n) = f(n)*A5822(n) for some f(n) that has been
>>> lost,
>>> >> and that this relation is only intended to hold for n>=7, or maybe
>>> n>=8.
>>> >>
>>> >> The cited works of Plouffe probably contain the explanation, but my
>>> >> French is not up to it.
>>> >>
>>> >> Brendan.
>>> >>
>>> >> --
>>> >> Seqfan Mailing list - http://list.seqfan.eu/
>>> >>
>>> > --
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>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>