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%I #58 Feb 21 2023 17:44:39
%S 0,1,3,7,11,15,21,31,39,43,49,61,77,91,105,127,143,147,153,165,181,
%T 197,217,249,285,307,321,349,391,431,467,517,549,553,559,571,587,603,
%U 623,655,691,715
%N H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).
%C An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.
%C The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks" with length sqrt(2).
%C Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.
%C Each hexagon has area = 4.
%C The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.
%C The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.
%C [It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - _N. J. A. Sloane_, Feb 06 2023]
%C From _Omar E. Pol_, Feb 17 2023: (Start)
%C This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.
%C The rules are as follows:
%C - The elements (toothpicks and D-toothpicks) are connected at their ends.
%C - At each free end of the elements of the old generation two elements of the new generation must be connected.
%C - The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.
%C - The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.
%C As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).
%C Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:
%C - Regular hexagons of area 4.
%C - Concave decagons (or concave 10-gons) of area 8.
%C - Concave dodecagons (or concave 12-gons) of area 12.
%C There are infinitely many of these polygons.
%C The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.
%C The structure has internal growth as some members of the mentioned family. (End)
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H Omar E. Pol, <a href="/A182838/a182838_1.jpg">Illustration of initial terms</a>
%H Omar E. Pol, <a href="/A182838/a182838_3.jpg">Illustration of a(32) = 549</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>
%F Conjecture: a(n) = (A182840(n+1) + A267458(n+1) - 2)/4. - _Omar E. Pol_, Feb 10 2023
%e We start at stage 0 with no toothpicks.
%e At stage 1 we place a D-toothpick [(0,1),(1,2)], so a(1)=1.
%e At stage 2 we place a toothpick [(1,2),(1,3)] and a D-toothpick [(1,2),(2,1)], so a(2)=1+2=3.
%e At stage 3 we place 4 elements: a D-toothpick [(1,3),(0,4)], a D-toothpick [(1,3),(2,4)], a D-toothpick [(2,1),(3,2)] and a toothpick [(2,1),(2,0)], so a(3)=3+4=7. Etc.
%e The first hexagon appears in the structure after 4 stages.
%Y Cf. A139250, A153000, A161206, A170888, A172308, A182632, A182634, A182839, A182840, A187212, A194444, A220524, A233970, A267458, A267694, A267698.
%Y See A360501 and A360512 for another hexagonal net built on the square grid. - _N. J. A. Sloane_, Feb 09 2023
%K nonn,more
%O 0,3
%A _Omar E. Pol_, Dec 12 2010
%E Partially edited by _N. J. A. Sloane_, Feb 06 2023
%E a(19)-a(41) from _Omar E. Pol_, Feb 06 2023
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