A173456 -id:A173456 - cp's OEIS frontend

A173457 Number of cell turned "ON" at n-th stage of cellular automaton of A173456.

0, 1, 8, 12, 4, 28, 36, 4, 28, 36, 12, 84, 108, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 12, 84, 108, 36, 252, 324, 36, 252, 324, 108, 756, 972, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324

Offset: 0

Omar E. Pol, Feb 18 2010

Essentially the first differences of A173456.

It appears that row lengths give A098011. After the initial zero, it appears that row lengths give the absolute values of A110164. - Omar E. Pol, Apr 22 2013

			From Omar E. Pol, Apr 22 2013 (Start):
When written as an irregular triangle begins:
0;
1;
8,12;
4,28,36;
4,28,36,12,84,108;
4,28,36,12,84,108,12,84,108,36,252,324;
4,28,36,12,84,108,12,84,108,36,252,324,12,84,108,36,252,324,36,252,324,108,756,972;
4,28,36,12,84,108,12,84,108,36,252,324,...
(End)

Lars Blomberg, Table of n, a(n) for n = 0..6000
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. A139250, A139251, A173456, A173458, A173459, A173461.

a(0)=0, a(1)=1, a(2)=8, a(3)=12, for n>=4 when (n MOD 3)=0,1,2 let m=36,4,28 then a(n)=m*A147610((n + 2) / 3). (Found empirically) [Lars Blomberg, Apr 22 2013]

a(41)-a(60) from Lars Blomberg, Apr 22 2013

A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 21, 29, 81, 93, 105, 189, 225, 253, 441, 453, 465, 549, 585, 621, 873, 909, 945, 1197, 1305, 1397, 2025, 2037, 2049, 2133, 2169, 2205, 2457, 2493, 2529, 2781, 2889, 2997, 3753, 3789, 3825, 4077, 4185, 4293, 5049, 5157, 5265, 6021, 6345, 6637, 8649

Offset: 0

Omar E. Pol, Feb 18 2010

nonn

On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.
In order to construct this sequence we use the following rules:
- If n is congruent to 0 (mod 3), we turn "ON" the cells around the vertex of every convex corner formed in the structure at the generation n-1. Note that every vertex is surrounded by three new "ON" cells.
- If n is congruent to 1 (mod 3), we turn "ON" the possible bridge cells and the possible peninsula cells (For the definition of bridge cell and of peninsula cell see A160117).
- If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.
- Everything that is already ON remains ON.
A173461, the first differences, gives the number of cells turned "ON" at n-th stage.

			Array begins:
0, 1, 9;
21, 29, 81;
93, 105, 189;
225, 253, 441;
453, 465, 549;
...
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
7..7.....7..7
.66.......66.
.65555555556.
7.545545545.7
..553353355..
..553222355..
..545212545..
..553222355..
..553353355..
7.545545545.7
.65555555556.
.66.......66.
7..7.....7..7

Lars Blomberg, Table of n, a(n) for n = 0..6000
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. A160117, A160118, A173456, A173461, A173462, A173463.

a(0)=0, a(n) = a(n-1) + A173461(n), n>=1. - [Lars Blomberg, Apr 23 2013]

a(18)-a(47) from Lars Blomberg, Apr 23 2013

A173458 a(n) = A173457(n+1)/2.

Original entry on oeis.org

4, 6, 2, 14, 18, 2, 14, 18, 6, 42, 54, 2, 14, 18, 6, 42, 54, 6, 42, 54, 18, 126, 162, 2, 14, 18, 6, 42, 54, 6, 42, 54, 18, 126, 162, 6, 42, 54, 18, 126, 162, 18, 126, 162, 54, 378, 486, 2, 14, 18, 6, 42, 54, 6, 42, 54, 18, 126, 162, 6, 42, 54, 18, 126, 162

Offset: 1

Omar E. Pol, Feb 18 2010

			From Omar E. Pol, Apr 22 2013 (Start):
When written as an irregular triangle begins:
4,6;
2,14,18;
2,14,18,6,42,54;
2,14,18,6,42,54,6,42,54,18,126,162;
2,14,18,6,42,54,6,42,54,18,126,162,6,42,54,18,126,162,18, 126,162,54,378,486;
2,14,18,6,42,54,6,42,54,18,126,162,6,42,54,18,126,162,...
(End)

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. A173456, A173457, A173459, A173462.

a(13)-a(65) from Lars Blomberg, Apr 22 2013

A173459 a(n) = A173457(n+1)/4.

Original entry on oeis.org

2, 3, 1, 7, 9, 1, 7, 9, 3, 21, 27, 1, 7, 9, 3, 21, 27, 3, 21, 27, 9, 63, 81, 1, 7, 9, 3, 21, 27, 3, 21, 27, 9, 63, 81, 3, 21, 27, 9, 63, 81, 9, 63, 81, 27, 189, 243, 1, 7, 9, 3, 21, 27, 3, 21, 27, 9, 63, 81, 3, 21, 27, 9, 63, 81, 9, 63, 81, 27, 189, 243, 3

Offset: 1

Omar E. Pol, Feb 18 2010

			From Omar E. Pol, Apr 22 2013 (Start):
When written as an irregular triangle begins:
2,3;
1,7,9;
1,7,9,3,21,27;
1,7,9,3,21,27,3,21,27,9,63,81;
1,7,9,3,21,27,3,21,27,9,63,81,3,21,27,9,63,81,9,63,81,27,189,243;
1,7,9,3,21,27,3,21,27,9,63,81,3,21,27,9,63,81,9,63,81,27,189,243,3,...
(End)

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. A173456, A173457, A173458, A173463.

a(13)-a(72) from Lars Blomberg, Apr 22 2013

cp's OEIS Frontend

A173457 Number of cell turned "ON" at n-th stage of cellular automaton of A173456.

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A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A173458 a(n) = A173457(n+1)/2.

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A173459 a(n) = A173457(n+1)/4.

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