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%I #18 Jul 21 2017 00:46:08
%S 1,2,4,6,9,11,15,19,24,26,30,34,40,44,52,60,69,71,75,79,85,89,97,105,
%T 115,119,127,135,147,155,171,187,204,206,210,214,220,224,232,240,250,
%U 254,262,270,282,290,306,322,340,344,352,360,372,380,396,412,432,440,456,472,496
%N 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.
%C 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.
%C 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).
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp052.jpg">Illustration of the structure on the square grid after 32 stages</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e Illustration of initial terms:
%e ---------------------------------------------------------
%e . A001316 A006046 Diagram A233777 a(n)
%e ---------------------------------------------------------
%e . 0
%e . * 1 1
%e . 1 1 |
%e . * 1 2
%e . 2 3 / \
%e . * * 2 4
%e . 2 5 | |
%e . * * 2 6
%e . 4 9 / \ / \
%e . * * * 3 9
%e . 2 11 | |
%e . * * 2 11
%e . 4 15 / \ / \
%e . * * * * 4 15
%e . 4 19 | | | |
%e . * * * * 4 19
%e . 8 27 / \ / \ / \ / \
%e . * * * * * 5 24
%e . 2 29 | |
%e . * * 2 26
%e . 4 33 / \ / \
%e . * * * * 4 30
%e .
%Y Partial sums of A233777.
%Y Cf. A001316, A006046, A047999, A139250, A141614, A160120, A160128, A160425, A194270, A194440, A233774.
%K nonn
%O 0,2
%A _Omar E. Pol_, Dec 25 2013
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