|
%I #36 Feb 24 2023 07:01:38
%S 0,1,2,4,4,4,6,10,8,4,6,12,16,14,14,22,16,4,6,12,16,16,20,32,36,22,14,
%T 28,42,40,36,50,32,4,6,12,16,16,20,32,36,24
%N Number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A182838.
%C From _Omar E. Pol_, Feb 06 2023: (Start)
%C The "word" of this cellular automaton is "ab".
%C Apart from the initial zero the structure of the irregular triangle is as shown below:
%C a,b;
%C a,b;
%C a,b,a,b;
%C a,b,a,b,a,b,a,b;
%C a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
%C ...
%C Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).
%C Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).
%C An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
%C Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.
%C For further information about the word of cellular automata see A296612.
%C It appears that the right border of the irregular triangle gives the even powers of 2. (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 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) = (A182841(n+1) + A010673(n))/4, n >= 2. - _Omar E. Pol_, Feb 10 2023
%e From _Omar E. Pol_, Feb 06 2023: (Start)
%e The nonzero terms can write as an irregular triangle as shown below:
%e 1, 2;
%e 4, 4;
%e 4, 6, 10, 8;
%e 4, 6, 12, 16, 14, 14, 22, 16;
%e 4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32;
%e ...
%e (End)
%Y First differences of A182838.
%Y Cf. A139250, A139251, A161207, A182633, A182635, A182841.
%Y Cf. A000079, A011782, A296612.
%K nonn,tabf,more
%O 0,3
%A _Omar E. Pol_, Dec 12 2010
%E a(19)-a(41) from _Omar E. Pol_, Jan 06 2023
|
