A169786 Triangle read by rows: T(n,k) is number of cells that turn from OFF to ON at stage k of the growth of the obtuse triangle of hexagons described in the comment.

1, 1, 2, 1, 1, 2, 3, 5, 3, 3, 1, 1, 2, 3, 5, 5, 5, 9, 13, 7, 3, 5, 8, 6, 4, 1, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 15, 3, 5, 8, 10, 10, 14, 22, 18, 8, 8, 13, 10, 5, 1, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63

Offset

1,3

Comments

Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905, A170906.

Assume the hexagons are oriented so that each one has a pair of vertical edges.

We label the cells in the usual way by Eisenstein integers, complex numbers r+sw, where r,s in Z, w = exp(2 pi i / 3) (see Conway and Sloane, pp. 52-53).

Initially all cells are OFF.

For x >= 1, define a roughly triangular region B_x by declaring the cells {sw: s >= 1}, {r-w: r >= -1}, {x-1-i+iw: 0 <= i <= x-2}, {x-1-i+(i+1)w: 0 <= i <= x-3} to be permanently OFF.

In other words, B_x consists of 0 plus the cells {r+sw: 0 <= s <= x-3, 1 <= r <= x-s-2}.

At stage 1, the "corner" cell 0 is turned ON; thereafter, a cell in B_x is turned ON if it has exactly one ON neighbor. Once a cell is ON it stays ON.

T(n,k) is the number of cells in B_{2^n} that are turned from OFF to ON at stage k (1 <= k <= 2^n-1).

Row n has 2^n-1 terms.

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd ed., 1988, see pp. 52-53.

Links

Table of n, a(n) for n=1..89.

Example

Example: B_8:
.W W W
..W 6 W W
...W 5 5 W W
....W 4 X 4 W W
.....W 3 3 4 X W W
......W 2 X 4 X 6 W W
.......1 2 3 4 5 6 7 W
........W W W W W W W
W = permanently OFF, X = OFF, ON cells are labeled with the stage at which they turned ON.
Triangle begins:
1,
1, 2, 1,
1, 2, 3, 5, 3, 3, 1,
1, 2, 3, 5, 5, 5, 9, 13, 7, 3, 5, 8, 6, 4, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 15, 3, 5, 8, 10, 10, 14, 22, 18, 8, 8, 13, 10, 5, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 31, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 42, 16, 8, 13, 18, 20, 24, 36, 36, 20, 15, 21, 15, 6, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 33, 5, 9, 15, 19, 19, 27, 43, 43, 27, 27, 47, 67, 71, 75, 99, 91, 41, 21, 37, 51, 55, 71, 111, 127, 87, 59, 87, 125, 119, 119, 133, 63, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 46, 22, 14, 24, 34, 38, 46, 70, 86, 68, 46, 58, 88, 98, 98, 114, 92, 32, 8, 13, 18, 20, 24, 36, 44, 36, 28, 38, 58, 70, 74, 88, 88, 52, 23, 21, 31, 38, 44, 60, 64, 44, 30, 33, 21, 7, 1,
...
The rows converge to A169787.

Crossrefs

Sequence in context: A058713 A331564 A288250 * A191631 A025259 A212359

Adjacent sequences: A169783 A169784 A169785 * A169787 A169788 A169789

Keyword

nonn,tabf

Author

David Applegate and N. J. A. Sloane, May 12 2010

Status

approved