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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 30-60-90 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 (list; graph; refs; listen; history; text; internal format)
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 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).

The rows converge to A170905. The rows sums give A170907.

Row n contains 2n-1 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

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

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Last modified February 23 17:40 EST 2018. Contains 299584 sequences. (Running on oeis4.)