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A170906
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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.
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4
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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, 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, 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
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OFFSET
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1,3
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COMMENTS
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Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905.
Assume the hexagons are oriented so that each one has a pair of vertical edges.
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.
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.
T(n,k) is the number of cells that are turned from OFF to ON at stage k (1 <= k <= 2n-1).
Row n contains 2n-1 terms.
I wish I had a recurrence for this sequence!
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LINKS
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EXAMPLE
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Triangle begins:
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 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 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
...
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.
7
.6
6.5
.X.4
X.4.3
.4.X.2
4.3.2.1
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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