[seqfan] Re: definition of A002848
David Newman
davidsnewman at gmail.com
Tue Feb 9 18:49:38 CET 2010
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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> >
>
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