A160172 -id:A160172 - cp's OEIS frontend

A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

0, 1, 5, 13, 21, 33, 53, 69, 77, 101, 117

Offset: 0

Omar E. Pol, Nov 12 2017

It seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints in the structure, so the structure is finished.
A294963(n) gives the number of elements added to the structure at n-th stage.

Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500, A294963.

For other hybrid cellular automata, see A289840, A290220, A294020, A294980.

A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

Original entry on oeis.org

1, 5, 13, 21, 33, 49, 65, 73, 97, 105

Offset: 1

Omar E. Pol, Mar 20 2018

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).
The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.
A299771(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Very similar to A294962.
Cf. A299771, A296612.
Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500.
For other hybrid cellular automata, see A194270, A220500, A289840, A290220, A294020, A294980.

A160714 T-toothpick sequence starting at the corner of an infinite square, but with angle = Pi/4.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 12, 21, 22, 25, 28, 37, 40, 49, 58, 85, 112, 113

Offset: 0

Omar E. Pol, Jun 02 2009

The sequence gives the number of T-toothpicks after n rounds.

See A160172 for more information about T-toothpick sequences.

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, A160172, A160173.

A183126 Toothpick sequence with toothpicks connected by their endpoints.

Original entry on oeis.org

0, 1, 7, 23, 39, 79, 95, 135, 175, 287, 303, 343, 383, 495, 535, 647, 759, 1087, 1103, 1143, 1183, 1295, 1335, 1447, 1559, 1887, 1927, 2039, 2151, 2479, 2591, 2919, 3247, 4223, 4239, 4279, 4319, 4431, 4471, 4583, 4695

Offset: 0

Omar E. Pol, Mar 28 2011

nonn

On the infinite square grid we start with no toothpicks.
At stage 1 we place a single toothpick of length 1.
Rule: each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A183127 gives the number of toothpicks added at the n-th stage.

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, p. 31.
John W. Layman, Graphs of the toothpick configuration for generations 1-15
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A160172, A160410, A183004, A183127, A183148.

a[n_] := 7 + 4 (n - 2 + Sum[3^DigitCount[k, 2, 1], {k, n - 2}]); a[0] = 0; a[1] = 1; Array[a, 41, 0] (* Michael De Vlieger, Nov 02 2022 *)

From Nathaniel Johnston, Apr 06 2011: (Start)
a(n) = 7 + 4*(n-2 + Sum_{k=1..n-2}3^A000120(k)), n >= 2.
a(n) = 7 + 4*(n-2 + 3*A151920(n-3)), n >= 3.
a(1 + 2^n) = 2^(n+2)+4^(n+1)-1, n >= 0.
(End)

Terms a(0)-a(10) confirmed and terms a(11)-a(35) added by John W. Layman, Mar 30 2011

a(36)-a(40) from Nathaniel Johnston, Apr 06 2011

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.

A160724 a(n) = A160714(n)*3.

Original entry on oeis.org

0, 3, 6, 15, 18, 27, 36, 63, 66, 75, 84, 111, 120, 147, 174, 255, 336, 339

Offset: 0

Omar E. Pol, Jun 02 2009

The sequence gives the number of T-toothpicks in the structure after n rounds.

See A160172 and A160714 for more information about T-toothpick sequences.

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, A160172, A160173, A160714, A160726.

a(9)-a(17) from 3*A160714(n) by Jinyuan Wang, Mar 14 2020

A160726 a(n) = A160714(n)*4.

Original entry on oeis.org

0, 4, 8, 20, 24, 36, 48, 84, 88, 100, 112, 148, 160, 196, 232, 340, 448, 452

Offset: 0

Omar E. Pol, Jun 02 2009

The sequence gives the number of T-toothpicks in the structure after n rounds.

See A160172 and A160714 for more information about T-toothpick sequences.

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, A160172, A160173, A160714, A160716, A160724.

a(9)-a(17) from 4*A160714(n) by Jinyuan Wang, Mar 14 2020

A160716 a(n) = A160714(n)*2.

Original entry on oeis.org

0, 2, 4, 10, 12, 18, 24, 42, 44, 50, 56, 74, 80, 98, 116, 170, 224, 226

Offset: 0

Omar E. Pol, Jun 02 2009

The sequence gives the number of T-toothpicks in the structure after n rounds.

See A160172 and A160714 for more information about T-toothpick sequences.

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, A160172, A160173, A160714, A160724, A160726.

A262609 Divisors of 1728.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

Offset: 1

Omar E. Pol, Nov 20 2015

A000578(12) = 1728 is the cube of 12.
The number of divisors of 1728 is A000005(1728) = 28.
The sum of the divisors of 1728 is A000203(1728) = 5080.
The prime factorization of 1728 is 2^6 * 3^3.
1728 + 1 = A001235(1) = A011541(2) = 1729 is the Hardy-Ramanujan number.
Three examples related to cellular automata:
1728 is also the number of ON cells after 32 generations of the cellular automata A160239 and A253088.
1728 is also the total number of ON cells around the central ON cell after 24 generations of the cellular automata A160414 and A256530.
1728 is also the total number of ON cells around the central ON cell after 43 generations of the cellular automata A160172 and A255366.

			a(3) * a(26) = 3 * 576 = 1728.
a(4) * a(25) = 4 * 432 = 1728.
a(5) * a(24) = 6 * 288 = 1728.

N. J. A. Sloane, Illustration of A160239(32) = A246030(5) = 1728
Index entries for sequences related to divisors of numbers

Cf. A000005, A000203, A000578, A001235, A011541, A133029, A160172, A160239, A160414, A246030, A253088, A255366, A256530.

Mathematica
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Divisors[1728]
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PARI
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divisors(1728)
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Sage
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divisors(1728);
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cp's OEIS Frontend

A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

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A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

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A160714 T-toothpick sequence starting at the corner of an infinite square, but with angle = Pi/4.

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A183126 Toothpick sequence with toothpicks connected by their endpoints.

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

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A160724 a(n) = A160714(n)*3.

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A160726 a(n) = A160714(n)*4.

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A160716 a(n) = A160714(n)*2.

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A262609 Divisors of 1728.

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Mathematica

PARI

Sage