A183060 -id:A183060 - cp's OEIS frontend

A186410 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A183060 using cubes.

0, 1, 6, 11, 32, 37, 58, 79, 180, 185, 206, 227, 328, 349, 450, 551, 1052, 1057, 1078, 1099, 1200, 1221, 1322, 1423, 1924, 1945, 2046, 2147, 2648, 2749, 3250, 3751, 6252, 6257, 6278, 6299, 6400, 6421, 6522, 6623

Offset: 0

Omar E. Pol, Feb 21 2011

nonn

The sequence gives the total number of cells turned ON after n stages in a cellular automaton based on Z^3 lattice in the same way that A183060 is based on the Z^2 lattice. In general here each cell has six neighbors.

It appears that after 2^k stages the structure resembles a pyramid. For the first differences see A186411.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 32-33.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A139250, A147562, A151781, A183060.

a[n_] := n + (4/5) Sum[5^DigitCount[i, 2, 1], {i, n - 1}]; Array[a, 40, 0] (* Michael De Vlieger, Nov 02 2022 *)

From Nathaniel Johnston, Mar 14 2011: (Start)
a(n) = n + (4/5)*(Sum_{i=1..n-1} 5^A000120(i)).
a(2^n) = 2^n + (4/5)*(6^n - 1).
(End)

More terms from Nathaniel Johnston, Mar 14 2011

A183061 First differences of A183060.

Original entry on oeis.org

0, 1, 3, 3, 7, 3, 7, 7, 19, 3, 7, 7, 19, 7, 19, 19, 55, 3, 7, 7, 19, 7, 19, 19, 55, 7, 19, 19, 55, 19, 55, 55, 163, 3, 7, 7, 19, 7, 19, 19, 55, 7, 19, 19, 55, 19, 55, 55, 163, 7, 19, 19, 55, 19, 55, 55, 163, 19, 55, 55, 163, 55, 163, 163, 487, 3

Offset: 0

Omar E. Pol, Feb 20 2011

nonn

The sequence gives the number of cells turned "ON" at the n-th stage in the structure of A183060.

			If written as a triangle begins:
0,
1,
3,
3,7,
3,7,7,19,
3,7,7,19,7,19,19,55,
3,7,7,19,7,19,19,55,7,19,19,55,19,55,55,163,
It appears that row sums give A007582.
It appears that last terms of rows give A100702.

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
Index entries for sequences related to cellular automata

Cf. A007582, A100702, A139250, A139251, A147582, A147610, A183060.

a(n) = 1 + A147582(n)/2.

a(n) = 1 + 2*A147610(n).

A183148 Toothpick sequence on the semi-infinite square grid with toothpicks connected by their endpoints.

Original entry on oeis.org

0, 1, 4, 13, 22, 43, 52, 73, 94, 151, 160, 181, 202, 259, 280, 337, 394, 559, 568, 589, 610, 667, 688, 745, 802, 967, 988, 1045, 1102, 1267, 1324, 1489, 1654, 2143, 2152, 2173, 2194, 2251, 2272, 2329, 2386, 2551, 2572, 2629

Offset: 0

Omar E. Pol, Mar 28 2011, Apr 03 2011

nonn

On the semi-infinite square grid we start with no toothpicks.
At stage 1 we place a single toothpick of length 1 which has one of its endpoints on the straight line.
New generations of toothpicks are added according to these rules: each exposed endpoint of toothpicks of the old generation must be touched by the 3 endpoints of three toothpicks of the new generation. Effectively these three toothpicks look like a T-toothpick (see A160172). The straight line that delimits the square grid acts like an impenetrable "absorbing" boundary: toothpicks may touch this line with at most one of their endpoints; these endpoints are not "exposed."
The sequence gives the number of toothpicks in the toothpick structure after n-th stage. The first differences (A183149) give the number of toothpicks added at n-th stage.

			At stage 1 place an orthogonal toothpick with one of its endpoints on the infinite straight line, so a(1) = 1. There is only one exposed endpoint.
At stage 2 place 3 toothpicks such that the structure looks like a cross, so a(2) = 1+3 = 4. There are 3 exposed endpoints.
At stage 3 place 9 toothpicks, so a(3) = 4+9 = 13. There are 3 exposed endpoints.
At stage 4 place 9 toothpicks, so a(4) = 13+9 = 22. There are 7 exposed endpoints.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 31-32.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A147562, A151920, A183004, A183060, A183126, A183149.

s[n_] := 1 + 4 Sum[3^(DigitCount[k, 2, 1] - 1), {k, n - 1}]; {0}~Join~Array[3 (# + (s[#] - 1)/2) + 1 &, 43, 0] (* Michael De Vlieger, Nov 02 2022 *)

a(n) = 3*A183060(n-1) + 1.

cp's OEIS Frontend

A186410 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A183060 using cubes.

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A183061 First differences of A183060.

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A183148 Toothpick sequence on the semi-infinite square grid with toothpicks connected by their endpoints.

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