[seqfan] Re: definition of A002848

[franktaw at netscape.net]

I should have added that my additional values match those in the Nigel 
Martin paper, found by Alois.  I've computed two more values, which 
still match.

>From Alois' post:

Nigel Martin: Solving a conjecture of Sedlacek:
   maximal edge sets in the 3-uniform sumset hypergraphs
   Discrete Mathematics, Volume 125, Issues 1-3,
   15 February 1994, Pages 273-277

1, 2, 4, 6, 3, 10, 25, 12, 42, 8, 40, 204, 21, 135, 1002,
4228, 720, 5134, 29546, 4079, 35533, 3040, 28777, 281504,
20505, 212283, 2352469, 16907265, 1669221

Franklin T. Adams-Watters

-----Original Message-----
From: franktaw at netscape.net

I beg to differ.  I 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."

Using the following PARI program:

nxyz(v,t)={local(n,r,x2);r=0;
 if(t==0,return(1));
 for(i3=3*t,#v,
  n=v[i3];
  for(i1=1,i3-2,
   x2=n-v[i1];
   if(x2<=v[i1],break);
   for(i2=i1+1,i3-1,
    if(v[i2]>=x2,
     if(v[i2]==x2,
      r+=nxyz(vector(i3-3,k,v[if(k<i1,k,if(k<i2-1,k+1,k+2))]),t-1));
     break))));
 r}

a(n)=nxyz(vector(n,k,k),n\3-(n%12==6|n%12==9))

I match A002849 except for a(1) and a(2), mentioned above, and a(14)
for which I get 204 instead of 202; this exception was noted by Alois
Heinz in his post.

The sequence then continues with:

135, 1002, 4228, 720, 5134, 29546, 4079

A002848(n) is A002849(n) for n == 0,3,7,10 (mod 12), 0 for n=1, and
A002849(n)-A002849(n-1) otherwise.

Franklin T. Adams-Watters

-----Original Message-----
From: N. J. A. Sloane <njas at research.att.com>

Dear Sequence Fans,  I will rewrite the definitions
of A002848 and A002849 to give more information

The various guesses that were made here were not right!

Neil



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