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%I #15 Feb 24 2021 02:48:19
%S 1,1,2,1,1,2,2,2,1,1,2,2,4,1,2,1,1,2,2,4,2,2,3,3,1,1,2,2,4,2,4,5,4,1,
%T 2,1,1,2,2,4,2,4,6,6,1,2,3,3,1,1,2,2,4,2,4,6,8,1,2,3,5,3,3,1,1,2,2,4,
%U 2,4,6,8,2,2,3,5,5,3,5,4,1,1,2,2,4,2,4,6,8,2,4,5,6,7,6,6,4,1,2,1
%N Triangle read by rows: T(n,k) = number of cells that are turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.
%C Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905.
%C Assume the hexagons are oriented so that each one has a pair of vertical edges.
%C Consider the (30 deg., 60 deg., 90 deg.) triangle of hexagons with n hexagons along the short side, along the X-axis, 2n-1 hexagons along the hypotenuse and n hexagons separated by single edges along the middle side, along the Y-axis.
%C Initially all cells are OFF. At stage 1, the cell in the 60-degree corner is turned ON; thereafter, a cell is turned ON if it has exactly one ON neighbor in the triangle. Once a cell is ON it stays ON.
%C T(n,k) is the number of cells that are turned from OFF to ON at stage k (1 <= k <= 2n-1).
%C The rows converge to A170905. The rows sums give A170907.
%C Row n contains 2n-1 terms.
%C I wish I had a recurrence for this sequence!
%H N. J. A. Sloane, <a href="/A170906/b170906.txt">Table of n, a(n) for n = 1..16384</a> (Rows 1..128, flattened)
%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 Triangle begins:
%e 1
%e 1 2 1
%e 1 2 2 2 1
%e 1 2 2 4 1 2 1
%e 1 2 2 4 2 2 3 3 1
%e 1 2 2 4 2 4 5 4 1 2 1
%e 1 2 2 4 2 4 6 6 1 2 3 3 1
%e 1 2 2 4 2 4 6 8 1 2 3 5 3 3 1
%e 1 2 2 4 2 4 6 8 2 2 3 5 5 3 5 4 1
%e 1 2 2 4 2 4 6 8 2 4 5 6 7 6 6 4 1 2 1
%e ...
%e Row n = 4, [1 2 2 4 1 2 1], corresponds to the sequence of cells being turned ON shown in the following triangle (X denotes a cell that stays OFF). The hexagons have to be imagined.
%e 7
%e .6
%e 6.5
%e .X.4
%e X.4.3
%e .4.X.2
%e 4.3.2.1
%Y Cf. A151723, A151724, A170905, A170907, A169782 (partial sums across rows).
%K nonn,tabf
%O 1,3
%A _N. J. A. Sloane_, Jan 24 2010
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