# [seqfan] Re: definition of A002848

franktaw at netscape.net franktaw at netscape.net
Tue Feb 9 23:08:17 CET 2010

```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

-----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.

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

Dear Sequence Fans,  I will rewrite the definitions

The various guesses that were made here were not right!

Neil

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