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.
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
KEYWORD
nonn,tabf
AUTHOR
David Applegate and N. J. A. Sloane, May 12 2010
STATUS
approved