A170906 -id:A170906 - cp's OEIS frontend

A170907 Row sums in triangle A170906.

1, 4, 8, 13, 20, 28, 37, 47, 59, 73, 87, 101, 118, 137, 156, 176, 198, 223, 248, 271, 299, 328, 357, 386, 418, 454, 489, 522, 558, 598, 638, 678, 720, 766, 812, 858, 907, 956, 1004, 1048, 1104, 1161, 1217, 1268, 1325, 1386, 1446, 1505, 1567, 1635, 1703, 1765

Offset: 1

N. J. A. Sloane, Jan 24 2010

nonn

Total number of cells that get turned ON in the n-th 30-60-90 triangle of hexagons described in A170906.

I wish I had a recurrence for this sequence!

David Applegate, Table of n, a(n) for n = 1..5989
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

Cf. A170906, A169757, A169758, A163862 (values at powers of 2).

More terms from R. J. Mathar, Jan 28 2010

A169782 Take the triangle in A170906 and replace each row with its partial sums.

Original entry on oeis.org

1, 1, 3, 4, 1, 3, 5, 7, 8, 1, 3, 5, 9, 10, 12, 13, 1, 3, 5, 9, 11, 13, 16, 19, 20, 1, 3, 5, 9, 11, 15, 20, 24, 25, 27, 28, 1, 3, 5, 9, 11, 15, 21, 27, 28, 30, 33, 36, 37, 1, 3, 5, 9, 11, 15, 21, 29, 30, 32, 35, 40, 43, 46, 47, 1, 3, 5, 9, 11, 15, 21, 29, 31, 33, 36, 41, 46, 49, 54, 58, 59, 1

Offset: 1

N. J. A. Sloane, May 11 2010

Row n has 2n-1 terms.

			Triangle begins:
. 1,
. 1,3,4,
. 1,3,5,7,8,
. 1,3,5,9,10,12,13,
. 1,3,5,9,11,13,16,19,20,
. 1,3,5,9,11,15,20,24,25,27,28,
. 1,3,5,9,11,15,21,27,28,30,33,36,37,
. 1,3,5,9,11,15,21,29,30,32,35,40,43,46,47,
. 1,3,5,9,11,15,21,29,31,33,36,41,46,49,54,58,59,
. ...

N. J. A. Sloane, Table of n, a(n) for n = 1..16384 (Rows 1..128, flattened)

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.

Original entry on oeis.org

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

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

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.

			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.

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

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A170907 Row sums in triangle A170906.

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A169782 Take the triangle in A170906 and replace each row with its partial sums.

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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.

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