A160120 -id:A160120 - cp's OEIS frontend

A160121 First differences of A160120.

1, 3, 3, 9, 3, 9, 9, 21, 9, 9, 9, 21, 15, 21, 27, 51, 27, 9, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 81, 21, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135, 141, 171, 279, 231, 69, 9, 21, 15, 21, 27

Offset: 1

Omar E. Pol, May 02 2009

nonn

Number of Y-toothpicks added at n-th stage to the Y-toothpick structure of A160120.

For a simpler version, see A151710. - Omar E. Pol, Dec 18 2012

			Contribution from _Omar E. Pol_, Jun 18 2009: (Start)
May be written as a triangle:
1,
3,
3,
9,
3,9,
9,21,9,9,
9,21,15,21,27,51,27,9,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,81,21,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279,231,69;
Rows converge to A161326.
(End)
Contribution from _Omar E. Pol_, Dec 18 2012: (Start):
Also this sequence may be written as another triangle (according to the structure of triangle A151710):
1;
3;
3,  9;
3,  9,9,21;
9,  9,9,21,15,21,27,51;
27, 9,9,21,15,21,27,51,33,21,27,51,51,57,69,117;
81,21,9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279;
(End)

JungHwan Min, Table of n, a(n) for n = 1..5000
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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [_Omar E. Pol_, Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A151710, A160171, A160173, A160120, A160715, A161207, A161329, A161331, A161326, A161431, A147582.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 1, 3, in == 0 && Count[out, 1] >= 2, 2, in == 0 && Count[out, 1] == 1, 1, True, in]]; A160121[n_] := Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)
A160121[n_] := Count[CellularAutomaton[{13390417258775213635414055181254541831894674613399006361662885886563211940509571858857491972104491013971547937418035084866785430974106432144737472376143620, 4, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009

A161828 Number of rhombuses in the Y-toothpick structure of A160120 after n rounds.

Original entry on oeis.org

0, 0, 3, 3, 9, 9, 15, 21, 33, 39

Offset: 0

Omar E. Pol, Jun 21 2009

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, A160120, A160124, A161419, A161418, A161836.

A160715 Y-toothpick sequence (as A160120) but without internal propagation in the structure.

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 61, 70, 79, 100, 115, 136, 163, 214, 217, 226, 235, 256, 271, 292, 319, 370, 391, 412, 439, 490, 541, 598, 667, 784, 787

Offset: 0

Omar E. Pol, Jun 02 2009

Note that A160120(9)=67 but in this sequence a(9)=61.

See the entry A160120 for more information about the Y-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. A000079, A000225, A006516, A139250, A160120, A151710.

More terms from Omar E. Pol, May 29 2010

A173066 a(n) = A130665(n-1) - A160120(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 6, 36, 12, 12, 12, 18, 12, 18, 18, 48, 24, 30, 30, 60, 36, 60, 72, 198, 120, 108, 108, 114, 108, 114, 114, 144, 120, 126, 126, 156, 132, 156, 168, 294, 216, 210, 210

Offset: 1

Omar E. Pol, May 29 2010

nonn

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. A130665, A139250, A160120, A160715, A173067, A173068, A173451, A173452, A173453.

More terms from Nathaniel Johnston, Nov 15 2010

A173068 a(n) = A160120(n) - A160715(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 42, 42, 42, 42, 42, 42, 42, 42, 120

Offset: 0

Omar E. Pol, May 29 2010

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. A130665, A139250, A160120, A160715, A173066, A173067, A173451, A173452, A173453.

a(14)-a(33) from Robert Price, Jun 17 2019

A161430 Fractal triangle whose virtual skeleton is a polyedge as the Y-toothpick structure of A160120.

Original entry on oeis.org

0, 9, 33, 57, 123, 147

Offset: 0

Omar E. Pol, Jun 17 2009

Number of ON states after n generations of cellular automaton based on the infinite triangular grid.

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, A160120, A161341, A161342, A161343.

A161836 Number of concave-convex hexagons in the Y-toothpick structure of A160120 after n rounds.

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 3, 9, 15

Offset: 0

Omar E. Pol, Jun 21 2009

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, A160120, A160124, A161419, A161418, A161828, A161837.

A266532 Total number of Y-toothpicks after n-th stage in the "outward" version of the cellular automaton of A160120.

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 61, 70, 79, 100, 109, 130, 151, 196, 199, 208, 217, 238, 247, 268, 289, 334, 343, 364, 385, 430, 451, 496, 541, 634, 637, 646, 655, 676, 685, 706, 727, 772, 781, 802, 823, 868, 889, 934, 979, 1072, 1081, 1102, 1123, 1168, 1189, 1234, 1279, 1372, 1393, 1438, 1483, 1576, 1621, 1714, 1807, 1996, 1999, 2008, 2017

Offset: 0

David Applegate and Omar E. Pol, Jan 18 2016

nonn

For the connection with A160720 (the "outward" version of the Ulam-Warburton cellular automaton A147562) see formula section and A267700.
A266533 (the first differences) gives the number of Y-toothpicks added to the structure at n-th stage.
First differs from A160120 at a(9).
First differs from A160715 at a(13).

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 cellular automata
Index entries for sequences related to toothpick sequences

Cf. A139250, A147562, A159912, A160120, A160715, A160720, A266533, A267700, A267701.

Conjecture: a(n) = 1 + 3*(A160720(n) - 1)/4 = 1 + 3*A267700(n-1), n >= 1. This formula is correct! - N. J. A. Sloane, Jan 23 2016

a(n) = 1 + 3*(A159912(n) - n)/2, n >= 1. - Omar E. Pol, Jan 24 2016

A160157 a(n) = 2*A160120(n).

Original entry on oeis.org

0, 2, 8, 14, 32, 38, 56, 74, 116, 134, 152, 170, 212, 242, 284, 338, 440, 494, 512, 530, 572, 602, 644, 698, 800, 866, 908, 962, 1064, 1166, 1280, 1418, 1652, 1814, 1856, 1874, 1916, 1946, 1988, 2042, 2144, 2210, 2252, 2306, 2408, 2510, 2624, 2762

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

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, A160120.

More terms from Omar E. Pol, Jan 09 2014

A160167 Total number of single toothpicks after n-th stage in the Y-toothpick structure of A160120.

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 174, 201, 228, 255, 318, 363, 426, 507, 660, 741, 768, 795, 858, 903, 966, 1047, 1200, 1299, 1362, 1443, 1596, 1749, 1920, 2127, 2478, 2721, 2784, 2811, 2874, 2919, 2982, 3063, 3216, 3315, 3378, 3459, 3612, 3765, 3936, 4143, 4494, 4755, 4854, 4935

Offset: 0

Omar E. Pol, Jun 01 2009, Jun 09 2009

nonn

Also, replace the Y-toothpick with the "three-diamonds" symbol, so we have a new cellular automaton in which a(n) counts the total number of diamonds in the structure after the n-th stage, A160120 also gives the total number of "three-diamonds" symbols after the n-th stage, and A253770 gives the total number of triangular ON cells after the n-th stage. - Omar E. Pol, Feb 10 2015

			From _Omar E. Pol_, Feb 10 2015: (Start)
After one generation, also, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are six ON cells and three diamonds, so a(1) = 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. A139250, A160120, A160157, A253770.

a(n) = 3*A160120(n).

a(n) = 3*A160157(n)/2 = A253770(n)/2. - Omar E. Pol, Feb 10 2015

New name and more terms from Omar E. Pol, Feb 10 2015

cp's OEIS Frontend

A160121 First differences of A160120.

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A161828 Number of rhombuses in the Y-toothpick structure of A160120 after n rounds.

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A160715 Y-toothpick sequence (as A160120) but without internal propagation in the structure.

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A173066 a(n) = A130665(n-1) - A160120(n).

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A173068 a(n) = A160120(n) - A160715(n).

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A161430 Fractal triangle whose virtual skeleton is a polyedge as the Y-toothpick structure of A160120.

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A161836 Number of concave-convex hexagons in the Y-toothpick structure of A160120 after n rounds.

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A266532 Total number of Y-toothpicks after n-th stage in the "outward" version of the cellular automaton of A160120.

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A160157 a(n) = 2*A160120(n).

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A160167 Total number of single toothpicks after n-th stage in the Y-toothpick structure of A160120.

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