A151724 -id:A151724 - cp's OEIS frontend

A256537 First differences of corner sequence A256536 associated with A151723.

1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65, 9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129

Offset: 1

Omar E. Pol, Apr 02 2015

Number of cells turned ON at n-th stage in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.

For an animation see "The movie version" in Links section.

			Written as an irregular triangle in which the row lengths are the absolute values of the terms of A141531, the sequence begins:
  1;
  3;
  5;
  9, 9;
  9, 17, 25, 17;
  9, 17, 29, 37, 33, 41, 57, 33;
  9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65;
  9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129;
  ...
It appears that the right border gives A083318, whose representation in base 2 gives A000533.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000533, A083318, A141531, A151723, A151724, A169779, A170898, A256536.

a(1) = 1; a(2) = 3.
It appears that a(n) = 1 + (A151724(n) + A151724(n-1))/3, n >= 3.
It appears that a(n) = 1 + (A151723(n) - A151723(n-2))/3, n >= 3.
It appears that a(n) = 1 + 2*(A170898(n-2) + A170898(n-3)), n >= 3.
a(3) = 5.
It appears that a(n) = 1 + 2*(A169779(n-2) - A169779(n-4)), n >= 4.

A169778 a(n) = ceiling(A170905/2).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 13, 9, 4, 7, 12, 14, 10, 16, 16, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 4, 7, 12, 14, 14, 22, 34, 30, 14, 16, 28, 35, 25, 35, 32, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21

Offset: 0

N. J. A. Sloane, May 08 2010

The hexagon-based CA of A151723 has as symmetry group the dihedral group of order 12. Consider a one-twelfth slice; a(n) is the number of cells that are turned from OFF to ON at generation n.

			Illustration of generations 1 through 9:
.1
..2
...3
..4.4
.....5
..7.6.6
...7...7
..8.8.8.8
.........9
...
From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle from 1, the right border gives A011782 and row lengths give A011782.
1,
1,
1,2,
1,2,3,4,
1,2,3,5,5,4,7,8,
1,2,3,5,5,5,9,13,9,4,7,12,14,10,16,16,
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,29,17,4,7,12,14,14,22,34,30,14,16,28,35,25,35,32;
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,... (End)

Cf. A151723, A151724, A170905.

A170905(n) = 2a(n) except A170905(1)=1.

A182842 a(n) = A182841(n+2)/2.

Original entry on oeis.org

2, 4, 7, 8, 7, 12, 19, 16, 7, 12, 23, 32, 27, 28, 43, 32, 7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64, 7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128

Offset: 0

Omar E. Pol, Dec 11 2010

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle with row lengths A011782:
2;
4;
7, 8;
7, 12, 19, 16;
7, 12, 23, 32, 27, 28, 43, 32;
7, 12, 23, 32, 31, 40, 63, 72, 43, 28, 55, 84, 79, 72, 99, 64;
7, 12, 23, 32, 31, 40, 63, 72, 47, 40, 71, 112, 119, 112, 143, 152, 75, 28, 55, 84, 91, 108, 163, 204, 151, 88, 131, 204, 207, 180, 219, 128;
The right border gives the even powers of 2, at least up a(2^9-1).
(End)

Olaf Voß, Table of n, a(n) for n = 0..998
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
Olaf Voß, Toothpick structures on hexagonal net
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A151724, A182633, A182840, A182841.

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011

A253771 First differences of A253770.

Original entry on oeis.org

0, 6, 18, 18, 54, 18, 54, 54, 126, 54, 54, 54, 126, 90, 126, 162, 306, 162, 54, 54, 126, 90, 126, 162, 306, 198, 126, 162, 306, 306, 342, 414, 702, 486, 126, 54, 126, 90, 126, 162, 306, 198, 126, 162, 306, 306, 342, 414, 702, 522, 198, 162

Offset: 0

Omar E. Pol, Jan 13 2015

Number of cells turned "ON" at n-th stage of cellular automaton of A253770.

Also 6 times the number of Y-toothpicks added at n-th stage in the Y-toothpick structure of A160120.

			Positive terms can be written as a triangle in which row lengths is A011782 as shown below:
6;
18;
18,  54;
18,  54,54,126;
54,  54,54,126,90,126,162,306;
162, 54,54,126,90,126,162,306,198,126,162,306,306,342,414,702;
486,126,54,126,90,126,162,306,198,126,162,306,306,342,414,702,522,198,162,...

Cf. A139251, A147582, A151724, A160121, A161645, A182633, A250301, A253770.

a(n) = 6*A160121(n).

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.

A250301 First differences of A250300.

Original entry on oeis.org

0, 3, 3, 6, 12, 12, 6, 12, 18, 18, 18, 18, 36, 36, 12, 24, 30, 18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54, 18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108

Offset: 0

Omar E. Pol, Jan 17 2015

Number of cells turned "ON" at n-th stage of cellular automaton of A250300.

Compare A161645.

			Positive terms can be written as a triangle in which row lengths is A011782 as shown below:
3;
3;
6,  12;
12,  6, 12, 18;
18, 18, 18, 36, 36, 12, 24, 30;
18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54;
18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108, ...

Cf. A139251, A147582, A151724, A160121, A161645, A182633, A250300, A253771.

A256139 First differences of A256138.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 36, 36, 28, 52, 60, 4, 12, 20, 36, 36, 36, 68, 100, 68, 28, 52, 92, 108, 76, 124, 124, 4, 12, 20, 36, 36, 36, 68, 100, 68, 36, 68, 116, 148, 132, 164, 228, 132, 28, 52, 92, 108, 108, 172, 268, 236, 108, 124, 220, 276, 196, 276, 252

Offset: 0

Omar E. Pol, Mar 20 2015

Number of cells turned ON at n-th stage in the structure of A256138.

First differs from A169708 at a(11).

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,36,36,28,52,60;
4,12,20,36,36,36,68,100,68,28,52,92,108,76,124,124;
4,12,20,36,36,36,68,100,68,36,68,116,148,132,164,228,132,28,52,92,108,108,172,268,236,108,124,220,276,196,276,252;
...
It appears that the right border gives A173033.

Cf. A028399, A011782, A151723, A151724, A160721, A169708, A170898, A170905, A173033, A256138.

a(n) = 2*A151724(n+1)/3, n >= 1.

A322663 First differences of A322662 divided by 12.

Original entry on oeis.org

1, 1, 7, 1, 6, 11, 14, 3, 11, 14, 25, 5, 18, 21, 37, 4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70, 10, 42, 41, 81, 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, 18, 37, 64, 151, 41, 80, 126, 131, 29

Offset: 1

Bradley Klee, Dec 22 2018

nonn

Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.

			Written as a 2^k triangle:
1,
1, 7,
1, 6,  11, 14,
3, 11, 14, 25, 5,  18, 21, 37,
4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70,  10, 42, 41, 81,
4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...

Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.

HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
MoveSet2 =Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
Clear@Pts;Pts[0] = {{0, 0}};
Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet2],1]],{x_,1}:>x]]];
Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0,32],2,1]]

a(n) = (A322662(n)-A322662(n-1))/12.

cp's OEIS Frontend

A256537 First differences of corner sequence A256536 associated with A151723.

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A169778 a(n) = ceiling(A170905/2).

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A182842 a(n) = A182841(n+2)/2.

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A253771 First differences of A253770.

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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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A250301 First differences of A250300.

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A256139 First differences of A256138.

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A322663 First differences of A322662 divided by 12.

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