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%I #34 Feb 17 2015 00:14:12
%S 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,
%T 57,141,209,261,1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,
%U 85,57,141,209,269,57,141,217,333,289,597,785,845,1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85
%N Difference after n generations between the total number of single toothpicks in the I-toothpick structure of A160164 and the total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.
%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>
%F a(n) = A160164(n) - A147562(n).
%e Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e 0;
%e 1;
%e 1,5;
%e 1,5,9,21;
%e 1,5,9,21,9,29,49,77;
%e 1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261;
%e 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;
%e ...
%e 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.
%e Also, written the positive terms as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e 1;
%e 1;
%e 5,1;
%e 5,9,21,1;
%e 5,9,21,9,29,49,77,1;
%e 5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,1;
%e 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;
%e ...
%e 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.
%e For example: after 8 generations the structures look like this:
%e .
%e . O
%e . O O O
%e . O O O
%e . _ _ _ _ _ _ _ _ O O O O O O O
%e . |_ _| |_ _| O O O O O
%e . | |_|_ _|_| | O O O O O O O O O
%e |_|_|_ _|_|_| O O O O O O O
%e . | | | | | O O O O O O O O O O O O O O O
%e . |_ _|_|_|_ _| O O O O O O O
%e . | |_|_ _|_| | O O O O O O O O O
%e . |_|_| |_|_| O O O O O
%e . _|_ _|_ _|_ _|_ O O O O O O O
%e . O O O
%e . 86 toothpicks O O O
%e . O
%e .
%e . 85 ON cells
%e .
%e 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.
%Y Cf. A126645, A139250, A147562, A160164, A169707, A170903.
%K nonn,tabf
%O 0,4
%A _Omar E. Pol_, Feb 15 2015
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