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EXAMPLE
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Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,5;
1,5,9,21;
1,5,9,21,9,29,49,77;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845;
...
It appears that the right border gives [0, 1] together with A126645. The right border gives the largest difference between both C.A. in every period.
Also, written the positive terms as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
1;
5,1;
5,9,21,1;
5,9,21,9,29,49,77,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845,1;
...
The right border gives A000012 according with the illustrations as shown below. In this triangle the right border gives the smallest difference between both C.A. in every period.
For example: after 8 generations the structures look like this:
.
. O
. O O O
. O O O
. _ _ _ _ _ _ _ _ O O O O O O O
. |_ _| |_ _| O O O O O
. | |_|_ _|_| | O O O O O O O O O
|_|_|_ _|_|_| O O O O O O O
. | | | | | O O O O O O O O O O O O O O O
. |_ _|_|_|_ _| O O O O O O O
. | |_|_ _|_| | O O O O O O O O O
. |_|_| |_|_| O O O O O
. _|_ _|_ _|_ _|_ O O O O O O O
. O O O
. 86 toothpicks O O O
. O
.
. 85 ON cells
.
a(8) = 1 because the I-toothpick structure contains 86 single toothpicks and the "Ulam-Warburton" two-dimensional cellular automaton has 85 ON cells, so the difference of the number of elements between both structures is equal to 86 - 85 = 1.
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