[seqfan] (no subject)

[Richard Mathar]

~s hexagonal toothpicks A161206 and A161328 extended

i) More terms have been obtained for the sequence of V-toothpicks on
the hexagonal grid
http://research.att.com/~njas/sequences/A161206 :
0,1,3,7,13,21,31,43,57,69,81,99,123,153,183,211,241,261,273,291,317,351,393,
443,499,553,597,645,709,791,871,939,1005,1041,1053,1071,1097,1131,1173,1223,
1281,1339,1393,1459,1549,1663,1789,1911,2031,2133,2193,2241,2311,2405,2523,
2665,2825,2987,3123,3253,3413,3617,3815,3975,4117,4185,4197,4215,4241,4275,
4317,4367,4425,4483,4537,4603,4693,4807,4933,

Illustrations are in
http://www.strw.leidenuniv.nl/~mathar/progs/a161206_2.eps
http://www.strw.leidenuniv.nl/~mathar/progs/a161206_3.eps
http://www.strw.leidenuniv.nl/~mathar/progs/a161206_4.eps
..
http://www.strw.leidenuniv.nl/~mathar/progs/a161206_15.eps

Pieces most recently added are colored green.

The first differences A161207 are
1,2,4,6,8,10,12,14,12,12,18,24,30,30,28,30,20,12,18,26,34,42,50,56,54,44,48,64,
82,80,68,66,36,12,18,26,34,42,50,58,58,54,66,90,114,126,122,120,102,60,48,70,
94,118,142,160,162,136,130,160,204,198,160,142,68,12,18,26,34,42,50,58,58,54,
66,90,114,126,


ii) Updating the trident toothpicks in 
http://research.att.com/~njas/sequences/A161328 needs
the special rule that the trident toothpicks are not added if this
would lead to overlap with another toothpick branch in the same generation.
(That is: a trident is not glued to an exposed point if any one of the
3 pieces of the trident would shortcut two exposed points.)
In these structures, "exposed" ends persist in generations:

0,1,4,9,16,29,40,57,72,93,116,141,168,201,228,253,268,293,328,369,424,477,536,
597,656,721,784,841,888,925,972,1037,1108,1205,1300,1405,1500,1589,1672,1753,
1840,1933,2012,2085,2164,2253,2360,2473,2592,2705,2820,2937,3072,3197,3324,
3453,3588,3741,3876,

Illustrations are in
http://www.strw.leidenuniv.nl/~mathar/progs/a161328_2.eps
http://www.strw.leidenuniv.nl/~mathar/progs/a161328_3.eps
http://www.strw.leidenuniv.nl/~mathar/progs/a161328_4.eps
..
http://www.strw.leidenuniv.nl/~mathar/progs/a161328_15.eps

Note that I get 72, not 76, for A161328(8). [Where would the additional
4 tridents be added in a161328_8.eps ? Which special rule did the author
of the sequence actually have in mind?]

First differences are A161329 (again different from the current sequence at
A161329(8)):
1,3,5,7,13,11,17,15,21,23,25,27,33,27,25,15,25,35,41,55,53,59,61,59,65,63,57,
47,37,47,65,71,97,95,105,95,89,83,81,87,93,79,73,79,89,107,113,119,113,115,117,
135,125,127,129,135,153,135


Richard Mathar