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%I #20 Feb 24 2021 02:48:18
%S 1,1,2,2,2,2,4,4,2,2,4,4,4,6,10,8,2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,
%T 16,2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,
%U 20,28,30,42
%N First differences of toothpick numbers A160406.
%C Number of toothpicks added at n-th stage in the toothpick structure of A160406.
%C From _Omar E. Pol_, Mar 15 2020: (Start)
%C The cellular automaton described in A160406 has word "ab", so the structure of this triangle is as follows:
%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 The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...
%C This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the even-indexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick.
%C For further information about the "word" of a cellular automaton see A296612. (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>
%e From _Omar E. Pol_, Jul 18 2009, Mar 15 2020: (Start)
%e If written as a triangle:
%e 1,1;
%e 2,2;
%e 2,2,4,4;
%e 2,2,4,4,4,6,10,8;
%e 2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16;
%e 2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,20,28,30,42;...
%e (End)
%Y Cf. A011782, A139250, A139251, A153000, A153006, A152980, A160406, A161830, A161831, A296612.
%K nonn
%O 1,3
%A _Omar E. Pol_, May 23 2009
%E More terms from _N. J. A. Sloane_, Jul 17 2009
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