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A255166 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.

%I A255166 #34 Feb 17 2015 00:14:12
%S A255166 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 A255166 57,141,209,261,1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,
%U A255166 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 A255166 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 A255166 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 A255166 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A255166 a(n) = A160164(n) - A147562(n).
%e A255166 Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e A255166 0;
%e A255166 1;
%e A255166 1,5;
%e A255166 1,5,9,21;
%e A255166 1,5,9,21,9,29,49,77;
%e A255166 1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261;
%e A255166 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 A255166 ...
%e A255166 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 A255166 Also, written the positive terms as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e A255166 1;
%e A255166 1;
%e A255166 5,1;
%e A255166 5,9,21,1;
%e A255166 5,9,21,9,29,49,77,1;
%e A255166 5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,1;
%e A255166 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 A255166 ...
%e A255166 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 A255166 For example: after 8 generations the structures look like this:
%e A255166 .
%e A255166 .                                      O
%e A255166 .                                    O O O
%e A255166 .                                  O   O   O
%e A255166 .    _ _ _ _ _ _ _ _             O O O O O O O
%e A255166 .     |_ _|   |_ _|            O   O   O   O   O
%e A255166 .     | |_|_ _|_| |          O O O   O O O   O O O
%e A255166       |_|_|_ _|_|_|        O   O   O   O   O   O   O
%e A255166 .     |   | | |   |      O O O O O O O O O O O O O O O
%e A255166 .     |_ _|_|_|_ _|        O   O   O   O   O   O   O
%e A255166 .     | |_|_ _|_| |          O O O   O O O   O O O
%e A255166 .     |_|_|   |_|_|            O   O   O   O   O
%e A255166 .    _|_ _|_ _|_ _|_             O O O O O O O
%e A255166 .                                  O   O   O
%e A255166 .     86 toothpicks                  O O O
%e A255166 .                                      O
%e A255166 .
%e A255166 .                                 85 ON cells
%e A255166 .
%e A255166 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 A255166 Cf. A126645, A139250, A147562, A160164, A169707, A170903.
%K A255166 nonn,tabf
%O A255166 0,4
%A A255166 _Omar E. Pol_, Feb 15 2015