[seqfan] Graphs for Heinz Conjecture on A21638

[Brad Klee]

Hi All,

I was reading about the upcoming Riordan prize and decided to take a look
at N.J.Sloane's example conjecture A216368. The connection to core sequence
A000081 is interesting.

I searched around and didn't find anything about this conjecture in the
mailing list archives, and I'm not sure if it's been a topic of discussion
on the list.

I know it's supposed to be a competition, but I hope you all don't mind if
I share the results of some new calculations.

I made a descendant graph that shows how each derivative relates to the
next:

http://imgur.com/NYdQLXW

In this picture the number of nodes in each row "k" gives the value a(n,k).

You can see at n=6 collisions begin to occur, and at n=7 the situation is
even worse.

I then computed a new triangle with collisions removed:

http://imgur.com/sBD0ha1

And values:

1
1 1
1 2 2
1 3 4 4
1 4 7 9 9
1 5 11 88 20 20
1 6 16 34 46 48 48

I suspect that these graphs will admit a recursive construction more
readily than the graphs with collisions. You can already see that each
graph "n-1" is a subgraph of graph "n" and that the remainder of graph "n"
is similar to graph "n-1" with additional branches.

If you seqfans think these graphs are helpful, I am not opposed to having
the included in the file for A216368.

Please send questions or comments.

Good luck with your investigations,

Brad