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A233776
Number of grid points that are covered on the semi-infinite hexagonal grid after n-th stage in a cellular automaton in which the toothpicks are connected by their endpoints but the toothpicks placed in northwest or northeast direction are prohibited, starting with a(0) = 1.
1
1, 2, 4, 6, 9, 11, 15, 19, 24, 26, 30, 34, 40, 44, 52, 60, 69, 71, 75, 79, 85, 89, 97, 105, 115, 119, 127, 135, 147, 155, 171, 187, 204, 206, 210, 214, 220, 224, 232, 240, 250, 254, 262, 270, 282, 290, 306, 322, 340, 344, 352, 360, 372, 380, 396, 412, 432, 440, 456, 472, 496
OFFSET
0,2
COMMENTS
a(n) is also the number of grid points that are covered on the semi-infinite square grid after n-th stage in a D-toothpick structure in which the toothpicks placed in horizontal direction are prohibited, starting with a(0) = 1.
Note that both the structure mentioned in the above comment and the structure mentioned in the definition are essentially equivalents. Both structures are related to the Sierpinski triangle (see A047999). The number of edges in the structure after n-th stage is A006046(n). The number of edges added at n-th stage is A001316(n-1), for n >= 1 (see example).
EXAMPLE
Illustration of initial terms:
---------------------------------------------------------
. A001316 A006046 Diagram A233777 a(n)
---------------------------------------------------------
. 0
. * 1 1
. 1 1 |
. * 1 2
. 2 3 / \
. * * 2 4
. 2 5 | |
. * * 2 6
. 4 9 / \ / \
. * * * 3 9
. 2 11 | |
. * * 2 11
. 4 15 / \ / \
. * * * * 4 15
. 4 19 | | | |
. * * * * 4 19
. 8 27 / \ / \ / \ / \
. * * * * * 5 24
. 2 29 | |
. * * 2 26
. 4 33 / \ / \
. * * * * 4 30
.
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 25 2013
STATUS
approved