

A170906


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 306090 triangle of hexagons defined in Comments.


4



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

1,3


COMMENTS

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 Xaxis, 2n1 hexagons along the hypotenuse and n hexagons separated by single edges along the middle side, along the Yaxis.
Initially all cells are OFF. At stage 1, the cell in the 60degree 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 <= 2n1).
The rows converge to A170905. The rows sums give A170907.
Row n contains 2n1 terms.
I wish I had a recurrence for this sequence!


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..16384 (Rows 1..128, flattened)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS


EXAMPLE

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


CROSSREFS

Cf. A151723, A151724, A170905, A170907, A169782 (partial sums across rows).
Sequence in context: A060778 A096492 A053874 * A123245 A262611 A110535
Adjacent sequences: A170903 A170904 A170905 * A170907 A170908 A170909


KEYWORD

nonn,tabf


AUTHOR

N. J. A. Sloane, Jan 24 2010


STATUS

approved



