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"Letter A" toothpick sequence (see Comments for precise definition).
4

%I #46 Dec 07 2019 00:45:03

%S 0,1,3,5,9,15,21,27,39,53,65,71,83,97,113,131,163,197,217,223,235,249,

%T 265,283,315,349,373,391,423,461,505,567,659,741,777,783,795,809,825,

%U 843,875,909,933,951,983,1021,1065,1127,1219,1301,1341,1359,1391,1429,1473,1535,1627,1713,1773,1835,1931

%N "Letter A" toothpick sequence (see Comments for precise definition).

%C This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).

%C For the construction of the sequence the rules are as follows:

%C On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.

%C For the next n generations we have that:

%C At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.

%C If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.

%C If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.

%C If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.

%C If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.

%C The sequence gives the number of toothpicks and D-toothpicks after n stages.

%C A323647 (the first differences) gives the number of elements added at the n-th stage.

%C Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.

%C The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

%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/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

%F a(n) = 1 + A160730(n-1), n >= 1.

%F a(n) = 1 + 2*A168112(n-1), n >= 1.

%e After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.

%e Note that angle between both D-toothpicks is 90 degrees.

%e .

%e *

%e * *

%e * * * * *

%e * *

%e * *

%e .

%e After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.

%e .

%e *

%e * *

%e * * * * *

%e * *

%e * * * * * * * * * *

%e .

%Y Cf. A139250, A168112, A160730, A296612, A323647.

%Y For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

%K nonn

%O 0,3

%A _Omar E. Pol_, Mar 07 2019