[seqfan] Re: definition of A002848

Tue Feb 9 23:21:59 CET 2010

Nowakowski's variant to 4n
produced
http://www.research.att.com/~njas/sequences/A014635
To 4n+1 produce
http://www.research.att.com/~njas/sequences/A033585

All sequences are decreasing
Artur


Richard Guy pisze:
> Indeed, but I'm as confused, perhaps even more so,
> than the rest of you.  Neil must have constructed
> this (these?) from my papers.  For considerably
> more detail, see Richard Nowakowski's 1975 Calgary
> MSc thesis on the Langford-Skolem problem.  This
> sh'd've been pursued further and publicized ...
>
> I don't understand A002848 or A002849.  No
> example is given in either case.  My guess is
> that it shd read    Number of sol'ns of x+y=z
> from {1,2,...,3m}  instead of ...,n}.  But
> that still doesn't make clear what is going on.
>
> Another good place to start is my Unity of
> Combinatorics lecture:
>
> Richard K.~Guy, The unity of combinatorics,
> {\it Proc.\ 25th Iran.\ Math.\ Conf., Tehran}
> (1994), {\it Math.\ Appl.}, {\bf329} 129--159,
> Kluwer Acad.\ Publ., Dordrecht, 1995;
> {\it MR} {\bf96k}:05001.
>
> I'm not clear what Neil had in mind, but
> perhaps he himself will remember.  The problem
> was to partition the numbers from 1 to 3m
> into  m  triples which each satisfy the
> equation  x + y = z.  There are numerous
> connexions with other combinatorial objects.
> As someone has pointed out, there are
> solutions only if  m == 0 or 1  mod 4.
>
> For  m = 1, the unique solution is  1 + 2 = 3.
>
> For  m = 4, there are 8 solutions:
>
> 1 5  6     1 5  6     2  5  7     1  6  7
> 2 8 10     3 7 10     3  6  9     4  5  9
> 4 7 11     2 9 11     1 10 11     3  8 11
> 3 9 12     4 8 12     4  8 12     2 10 12
>
> 2 4  6     2  6  8    3  4  7     3  5  8 
> 1 9 10     4  5  9    1  8  9     2  7  9
> 3 8 11     3  7 10    5  6 11     4  6 10
> 5 7 12     1 11 12    2 10 12     1 11 12
>
> I don't think that we found all solutions
> for any larger values of  m.  Computers
> and computation have advanced quite a bit
> in the last 36 years, and some of you can
> probably push this quite a way.  Nowakowski
> gives (some) solutions for  m = 4k  and
> 4k+1  for all  k.
>
> But I don't think this answers your questions.
>
> Come in, Neil!      R.
>
>
>
> On Tue, 9 Feb 2010, David Newman wrote:
>
>   
>> Pardon my ignorance, but is there an option of writing to Guy and asking
>> him?
>>
>> On Tue, Feb 9, 2010 at 12:09 PM, <franktaw at netscape.net> wrote:
>>
>>     
>>> Possibly (I haven't really checked, but the pattern is right) A002849
>>> is the number of partitions of a subset of 1..n into triples X+Y=Z,
>>> with the maximum possible number of such triples.  A002848 would then
>>> be the number of such partitions that include n in one of the triples.
>>>
>>> If this is correct, I would argue that A002849(1) and A002849(2) should
>>> both be 1, representing the empty partition.
>>>
>>> Franklin T. Adams-Watters
>>>
>>> -----Original Message-----
>>>  From: Andrew Weimholt <andrew.weimholt at gmail.com>
>>>
>>> A002849 has the same definition, but different terms. One or both
>>> definitions are wrong.
>>>
>>> If you google "unity of combinatorics", the third result is a google
>>> books page.
>>> R. K. Guy briefly discusses the X+Y=Z problem and the X+Y=2Z problem.
>>>
>>> Given the numbers 1 to 3n, the goal is to partition them into triples
>>> such that
>>> each triple is a solution to X+Y=Z. For example...
>>>
>>> 1+11=12,
>>> 2+6=8,
>>> 3+7=10,
>>> 4+5=9
>>>
>>> is one solution for n=4
>>>
>>> here are two more for n=4
>>>
>>> 1+11=12
>>> 2+7=9
>>> 3+5=8
>>> 4+6=10
>>>
>>> 1+5=6
>>> 2+8=10
>>> 3+9=12
>>> 4+7=11
>>>
>>> Unfortunately, this still doesn't shed much light on A002848 and
>>> A002849, as the terms
>>> do not seem to match.
>>>
>>> As Guy notes, the X+Y=Z problem only has solutions for n == 0 or 1 mod
>>> 4,
>>> whereas A002848 and A002849 only contain zeros for a(1) and a(2).
>>>
>>> Something else is still missing.
>>>
>>> Andrew
>>>
>>>
>>> On Mon, Feb 8, 2010 at 11:03 PM,  <franktaw at netscape.net> wrote:
>>>       
>>>> Nothing so straightforward is going to work, because the sequence
>>>>         
>>> terms
>>>       
>>>> as they exist are not monotonic.  The first reference, "R. K. Guy,
>>>> ``Sedlacek's Conjecture on Disjoint Solutions of x+y= z,'' in Proc.
>>>> Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.", looks to me
>>>>         
>>> like
>>>       
>>>> the place to start; unfortunately, I don't have access to this.
>>>>
>>>> Franklin T. Adams-Watters
>>>>
>>>> -----Original Message-----
>>>> From: Rainer Rosenthal <r.rosenthal at web.de>
>>>>
>>>> Max Alekseyev wrote:
>>>>         
>>>>> Does anybody understand the definition of A002848 and how it produces
>>>>> the listed terms?
>>>>>           
>>>> Just a guess, to be verified:  "number of solutions of x+y+z=n"
>>>>
>>>> a(3)=1 solutions: 1+1+1=3
>>>> a(4)=1 solutions: 1+1+2=4
>>>> a(5)=2 solutions: 1+1+3=5 and 1+2+2=5
>>>> a(6)=2 solutions: 1+1+4=6 and 1+2+3=6
>>>>
>>>> oops ... there is 2+2+2=6 as well.
>>>> What a pity :-(
>>>>
>>>>
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