Multiply Then Add = Primes, Composites

[Leroy Quet]

sets {a(k)} and {b(k)}.
same side as 6, or else 2 divides S. And finally, 7 must be on the same 
side as 14, or else 7 divides S.
sequence defined below:
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Date: Fri, 5 Jan 2007 12:44:33 -0800
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "Sequence Fans" <seqfan at ext.jussieu.fr>, jvospost2 at yahoo.com,
   andrewpost at gmail.com
Subject: Re: Extending A066951 to a(7) = 70
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Related to A066951 Number of nonisomorphic connected graphs that can be
drawn in the plane with n unit-length edges; and A003055 Number of connected
graphs, up to homeomorphism, that can be drawn in the plane with n
unit-length edges:

New sequence (offset 1,3):

a(n) Number of connected graphs, up to diffeomorphism, that can be drawn in
the plane with n unit-length edges.

We allow nodes to be "hinged" for edges to rotate freely, up to edges not
being allowed to cross and nodes not being able to touch edges in between
nodes.  Hence we can't, in the plane, smoothly turn:

0--0--0
|  |
|  |
0--0

into

0--0
| /|
|0 |
0--0

Up until n=5 and the above example, a(n) = A0066951.

a(n) = 1,1,3,5,13,37,...

For n=6 we have 9 new shapes beyond those of A0066951(6):
(1) added to the original pentagon with an extra edge from a node to an
external node unit distance away, 1 new shape: the nondiffeomorphic pentagon
with an extra edge from a node to an internal node unit distance away;
(2) added to the original square with a 2-path hanging from a node
(externally) [ethyl cyclobutane], 1 new shape, the square with a 2-path
hanging from a node (internally), room being made by deforming square to
parallogram;
(3) added to the original square with two distinct edges to 2 nodes hanging
from a corner node (externally) [1,1-dimethylcyclobutane], 2 new
shapes, the2 nondiffeomorphic
variants: square with two distinct edges to 2 nodes hanging from a corner
node (1 externally, 1 internally) and square with two distinct edges to 2
nodes hanging from a corner node (2 internally);
(4) added to the original square with two distinct edges to 2 nodes hanging
from 2 adjacent corner nodes (externally) [cis-dimethylcyclobutane], 2 new
shapes, the 2 nondiffeomorphic variants: square with two distinct edges to 2
nodes hanging from 2 adjacent corner nodes (1 externally, 1
internally) and square
with two distinct edges to 2 nodes hanging from 2 adjacent corner nodes (2
internally);
(5) added to the original square with two distinct edges to 2 nodes hanging
from opposite corner nodes (externally) [trans-dimethylcyclobutane], 2 new
shapes, the 2 nondiffeomorphic variants: square with two distinct edges to 2
nodes hanging from opposite corner nodes (1 externally, 1 internally)
and square
with two distinct edges to 2 nodes hanging from opposite corner node (2
internally);
(6) added to the original square with an equilateral triangle sharing an
edge and 2 nodes with the square (externally), 1 new shape, the
nondiffeomorphic
variant square with an equilateral triangle sharing an edge and 2 nodes with
the square (internally).

I have drawings of many new shapes with n=7, but am not sure yet that my set
is complete.

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Related to 
A066951 Number of nonisomorphic connected graphs that can be drawn in the plane with n unit-length edges; and A003055 Number of connected graphs, up to homeomorphism, that can be drawn in the plane with n unit-length edges:
<br><br>New sequence (offset 1,3):<br><br>a(n) Number of connected graphs, up to diffeomorphism, that can be drawn in the plane with n unit-length edges.<br><br>We allow nodes to be "hinged" for edges to rotate freely, up to edges not being allowed to cross and nodes not being able to touch edges in between nodes.  Hence we can't, in the plane, smoothly turn:
<br><br><span style="font-family: courier new,monospace;">0--0--0</span><br style="font-family: courier new,monospace;"><span style="font-family: courier new,monospace;">|  |</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">|  |</span><br><span style="font-family: courier new,monospace;">0--0<br><br>into<br><br></span><span style="font-family: courier new,monospace;">0--0</span><br style="font-family: courier new,monospace;">

<span style="font-family: courier new,monospace;">| /|</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">|0 |</span><br>
<span style="font-family: courier new,monospace;">0--0<br><br>Up until n=5 and the above example, a(n) = A0066951.<br><br>a(n) = 1,1,3,5,13,37,...<br><br>For n=6 we have 9 new shapes beyond those of </span><span style="font-family: courier new,monospace;">
A0066951(6)</span><span style="font-family: courier new,monospace;">:<br>(1) added to the original pentagon with an extra edge from a node to an external node unit distance away, 1 new shape: the nondiffeomorphic </span><span style="font-family: courier new,monospace;">
pentagon with an extra edge from a node to an internal node unit distance away</span><span style="font-family: courier new,monospace;">;<br>(2) added to the original square with a 2-path hanging from a node (externally) [ethyl cyclobutane], 1 new shape, the 
</span><span style="font-family: courier new,monospace;">square with a 2-path hanging from a node (internally), room being made by deforming square to parallogram;<br>(3) </span><span style="font-family: courier new,monospace;">
added to the original square with two distinct edges to 2 nodes hanging from a corner node (externally) [1,1-dimethylcyclobutane], 2 new shapes, the</span><span style="font-family: courier new,monospace;"> 2 </span><span style="font-family: courier new,monospace;">
nondiffeomorphic variants: </span><span style="font-family: courier new,monospace;">square with two distinct edges to 2 nodes hanging from a corner node (1 externally, 1 internally) and </span><span style="font-family: courier new,monospace;">
square with two distinct edges to 2 nodes hanging from a corner node (2 internally);<br></span><span style="font-family: courier new,monospace;">(4) </span><span style="font-family: courier new,monospace;">added
to the original square with two distinct edges to 2 nodes hanging from 2 adjacent corner nodes (externally) [cis-dimethylcyclobutane], </span><span style="font-family: courier new,monospace;">2 new shapes,</span><span style="font-family: courier new,monospace;">
 the</span><span style="font-family: courier new,monospace;"> 2 </span><span style="font-family: courier new,monospace;">nondiffeomorphic variants: </span><span style="font-family: courier new,monospace;">square with two distinct edges to 2 nodes hanging from 2 adjacent corner nodes (1 externally, 1 internally) and 
</span><span style="font-family: courier new,monospace;">square with two distinct edges to 2 nodes hanging from 2 adjacent corner nodes (2 internally);<br></span><span style="font-family: courier new,monospace;">(5) </span>
<span style="font-family: courier new,monospace;">added
to the original square with two distinct edges to 2 nodes hanging from opposite corner nodes (externally) [trans-dimethylcyclobutane], </span><span style="font-family: courier new,monospace;">2 new shapes,</span><span style="font-family: courier new,monospace;">
 the</span><span style="font-family: courier new,monospace;"> 2 </span><span style="font-family: courier new,monospace;">nondiffeomorphic variants: </span><span style="font-family: courier new,monospace;">square with two distinct edges to 2 nodes hanging from opposite corner nodes (1 externally, 1 internally) and 
</span><span style="font-family: courier new,monospace;">square with two distinct edges to 2 nodes hanging from opposite corner node (2 internally);<br>(6) </span><span style="font-family: courier new,monospace;">added
to the original square with an equilateral triangle sharing an edge and 2 nodes with the square (externally), 1 new shape, the </span><span style="font-family: courier new,monospace;">nondiffeomorphic variant </span><span style="font-family: courier new,monospace;">
square with an equilateral triangle sharing an edge and 2 nodes with the square (internally).<br><br>I have drawings of many new shapes with n=7, but am not sure yet that my set is complete.<br></span><span style="font-family: courier new,monospace;">
<br></span>

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