[seqfan] a question about arrangements of pennies
N. J. A. Sloane
njas at research.att.com
Mon Dec 14 05:02:26 CET 2009
Dear Seq Fans, J. Lowell submitted a sequence today
which I am still stuggling with. His definition and terms
seemed unsatisfactory.
First let me state a simpler question.
1. Take the standard triangular lattice packing
of pennies, in which each penny touches 6 others.
How many ways are there to pick n pennies (modulo
rotations and reflections) such that if you construct
the graph with nodes = centers of pennies,
edges = pairs of touching pennies,
then this graph is connected and every edge
belongs to at least one triangle?
Here's what I get for the first few cases:
n 1 2 3 4 5 6 7
# 1 0 1 1 2 3 6
Here they are:
o
o o
o
o o
o
o o
o o o
o
o o o
o
I'll skip 6
o o o
o o o o
o o
o o o
o o
o o
o o
o o o
o o
o o o o
o
o o
o o o
o o
o
o o o o
o o
This may well be in the OEIS already. It reminds me of the problems Ron Graham and I looked at in this paper:
Penny-Packing and Two-Dimensional Codes, R. L. Graham and N. J. A. Sloane, Discrete and Comput. Geom., 5 (1990), pp. 1-11. (Available from my home page, Pub List, item 138)
2. J Lowell asked (in effect) how many of these arrangements of
pennies are "rigid", without defining the term.
This is not "rigid", for example,
o
o o o
o
because the central node can act as a hinge,
and one triangle can rotate with respect to the other.
There are several ways to enforce rigidity. Perhaps
the best is to add a further constraint to the first
problem. Form a second graph with nodes = triangles,
edges = two triangles that share an edge. Then this second graph
must be connected.
For this arrangement
o
o o o
o
the second graph contains 2 isolated nodes, so is ruled out.
Here are Lowell's values, as corrected by me:
%S A171604 1,0,1,1,1,3,4
Can anyone compute a few more terms of either sequence?
Neil
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