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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, 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.]

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 January 20 12:50 EST 2019. Contains 319330 sequences. (Running on oeis4.)